arXiv:2007.11384v1 [math.DG] 22 Jul 2020 · THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE...

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H 1 VALENTINA FRANCESCHI 1 , ROBERTO MONTI 2 , ALBERTO RIGHINI 2 , AND MARIO SIGALOTTI 1 Abstract. We study the isoperimetric problem for anisotropic left-invariant perime- ter measures on R 3 , endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm φ on the horizontal distribution. We first prove a representation formula for the φ-perimeter of regular sets and, assuming some regularity on φ and on its dual norm φ * , we deduce a foliation property by sub-Finsler geodesics of C 2 -smooth surfaces with constant φ-curvature. We then prove that the characteristic set of C 2 -smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize C 2 -smooth φ-isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. We also show, under suitable regularity properties on φ, that such sub-Finsler candi- date isoperimetric sets are indeed C 2 -smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where φ is crystalline). 1. Introduction Let φ : R n [0, ) be a norm in R n , n 2. The associated Finsler or anisotropic perimeter of a Lebesgue measurable set E R n is defined as P φ (E) = sup Z E div(V ) dp : V C c (R n ; R n ) with max pR n φ(V (p)) 1 . If E is regular, P φ (E) can be represented as a surface integral as follows P φ (E)= Z ∂E φ * (ν E ) dH n-1 , where ν E is the inner unit normal to ∂E and φ * : R n [0, ) is the dual norm defined by φ * (w) = max φ(v)=1 hw, vi, w R n . 2010 Mathematics Subject Classification. 49Q10, 52B60, 53C17. Key words and phrases. Isoperimetric problem, anisotropic perimeter, Heisenberg group, sub- Finsler geometry, Wulff shapes, Pansu’s bubbles. 1 arXiv:2007.11384v1 [math.DG] 22 Jul 2020

Transcript of arXiv:2007.11384v1 [math.DG] 22 Jul 2020 · THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE...

Page 1: arXiv:2007.11384v1 [math.DG] 22 Jul 2020 · THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 VALENTINA FRANCESCHI1, ROBERTO MONTI2, ALBERTO RIGHINI2, AND MARIO SIGALOTTI1

THE ISOPERIMETRIC PROBLEM FOR REGULAR ANDCRYSTALLINE NORMS IN H1

VALENTINA FRANCESCHI1, ROBERTO MONTI2, ALBERTO RIGHINI2,

AND MARIO SIGALOTTI1

Abstract. We study the isoperimetric problem for anisotropic left-invariant perime-

ter measures on R3, endowed with the Heisenberg group structure. The perimeter

is associated with a left-invariant norm φ on the horizontal distribution. We first

prove a representation formula for the φ-perimeter of regular sets and, assuming

some regularity on φ and on its dual norm φ∗, we deduce a foliation property by

sub-Finsler geodesics of C2-smooth surfaces with constant φ-curvature. We then

prove that the characteristic set of C2-smooth surfaces that are locally extremal for

the isoperimetric problem is made of isolated points and horizontal curves satisfying

a suitable differential equation. Based on such a characterization, we characterize

C2-smooth φ-isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. We

also show, under suitable regularity properties on φ, that such sub-Finsler candi-

date isoperimetric sets are indeed C2-smooth. By an approximation procedure, we

finally prove a conditional minimality property for the candidate solutions in the

general case (including the case where φ is crystalline).

1. Introduction

Let φ : Rn → [0,∞) be a norm in Rn, n ≥ 2. The associated Finsler or anisotropic

perimeter of a Lebesgue measurable set E ⊂ Rn is defined as

Pφ(E) = sup

{∫E

div(V ) dp : V ∈ C∞c (Rn;Rn) with maxp∈Rn

φ(V (p)) ≤ 1

}.

If E is regular, Pφ(E) can be represented as a surface integral as follows

Pφ(E) =

∫∂E

φ∗(νE) dH n−1,

where νE is the inner unit normal to ∂E and φ∗ : Rn → [0,∞) is the dual norm

defined by

φ∗(w) = maxφ(v)=1

〈w, v〉, w ∈ Rn.

2010 Mathematics Subject Classification. 49Q10, 52B60, 53C17.Key words and phrases. Isoperimetric problem, anisotropic perimeter, Heisenberg group, sub-

Finsler geometry, Wulff shapes, Pansu’s bubbles.1

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2 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Here, 〈·, ·〉 denotes the standard Euclidean scalar product in Rn and | · | the Euclidean

norm. In the theory of crystals, φ∗ is the surface tension of the interface between an

anisotropic material E and a fluid, and Pφ(E) is the total free energy.

In the case where φ = | · |, Pφ is the standard De Giorgi perimeter and isoperimetric

sets (i.e., sets of fixed volume that minimize perimeter) are Euclidean balls. For a

general norm φ, isoperimetric sets are translations and dilations of the Wulff shape,

first considered by G. Wulff in [31]. In our notation it corresponds to the unit ball of

the φ-norm. The first complete proof of the isoperimetric property of Wulff shapes in

the class of Lebesgue measurable sets with given volume is contained in [11, 12], and

based on the Brunn-Minkowski inequality. We refer to [10] for a quantitative version.

In this paper, we study the isoperimetric problem for sub-Finsler perimeter mea-

sures in the Heisenberg group H1. The latter is R3 endowed with the non-commutative

group law

(ξ, z) ∗ (ξ′, z′) = (ξ + ξ′, z + z′ + ω(ξ, ξ′)) , ξ, ξ′ ∈ R2, z, z′ ∈ R, (1.1)

where ω : R2 × R2 → R is the symplectic form

ω(ξ, ξ′) =1

2(xy′ − x′y), ξ = (x, y), ξ′ = (x′, y′) ∈ R2. (1.2)

The vector fields

X =∂

∂x− y

2

∂zand Y =

∂y+x

2

∂z

are left-invariant for the group action and span a two-dimensional distribution D(H1)

in TH1, called the horizontal distribution. We denote by D(p) the fiber of D at p ∈ H1.

Given a norm φ : R2 → [0,∞), the associated anisotropic perimeter measure in H1

is introduced in Definition 2.1 and takes into account only horizontal directions. For

a regular set E ⊂ R3 it can be represented as

Pφ(E) =

∫∂E

φ∗(NE) dH 2,

where NE is obtained by projecting the inner unit normal νE onto the horizontal

distribution. A set E ⊂ H1 is said to be φ-isoperimetric if there exists m > 0 such

that E minimizes

inf{Pφ(E) : E ⊂ H1 measurable, L 3(E) = m

}. (1.3)

If φ = | · | is the Euclidean norm in R2, then Pφ corresponds to the standard

horizontal perimeter in H1, introduced and studied in [7, 17, 16]. In this case, the

problem of characterizing φ-isoperimetric sets in the class of Lebesgue measurable

sets in H1 is open. According to Pansu’s conjecture [24], | · |-isoperimetric sets are

obtained through left-translations and anisotropic dilations δλ : H1 → H1, λ > 0,

δλ(ξ, z) = (λξ, λ2z), (1.4)

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 3

of the so-called Pansu’s bubble.

An absolutely continuous curve γ : I → H1 is said to be horizontal if γ(t) ∈ D(γ(t))

for a.e. t ∈ I and we call horizontal lift of an absolutely continuous curve ξ : I → R2

any horizontal curve γ = (ξ, z) with

z = ω(ξ, ξ).

Pansu’s bubble is the bounded set whose boundary is foliated by horizontal lifts of

planar circles of a given radius, passing through the origin. Such horizontal curves are

length minimizing between their endpoints for the sub-Riemannian distance in H1,

so that Pansu’s conjecture in H1 explicits a relation between isoperimetric sets and

the geometry of the ambient space. The conjecture is supported by several results

contained in [27, 22, 23, 26, 15, 14], but it is still unsolved in its full generality. A

quantitative version of the Heisenberg isoperimetric inequality has been proposed in

[13].

Very little is known on the isoperimetric problem when φ : R2 → [0,∞) is a general

norm in R2. While preparing the final version of this article, we became aware that

J. Pozuelo and M. Ritore have recently obtained several results on the problem,

considering also the case where φ is convex and homogeneous, but not necessarily a

norm (see [25]).

Existence of φ-isoperimetric sets follows by the arguments of [19], see Section 3.

The construction of the Pansu’s bubble can be generalized to the sub-Finsler context

in the following way. We call φ-circle of radius r > 0 and center ξ0 ∈ R2 the set

Cφ(ξ0, r) = {ξ ∈ R2 : φ(ξ − ξ0) = r}, (1.5)

and we call φ-bubble the bounded set Eφ whose boundary is foliated by horizontal

lifts of φ-circles in the plane of a given radius, passing through the origin.

In Figure 1 we represent two φ-bubbles, corresponding to φ = `p, with `p(x, y) =

(|x|p + |y|p)1p , in the cases p = 3 and p = 100. The latter can be seen as an approxi-

mation of the crystalline case.

Our main result is the characterization of C2-smooth φ-isoperimetric sets when φ

and φ∗ are C2-smooth. This result suggests that the φ-bubble is the solution to the

isoperimetric problem for Pφ. Here and in the following, if φ ∈ Ck(R2 \ {0}) we say

that φ is of class Ck, for k ∈ N.

Theorem 1.1. Let φ be a norm of class C2 such that φ∗ is of class C2. If E ⊂ H1

is a φ-isoperimetric set of class C2 then we have E = Eφ, up to left-translations and

anisotropic dilations.

The proof of Theorem 1.1 is presented in Section 8 and is based on a fine study of

the characteristic set of isoperimetric sets. The characteristic set of a set E ⊂ H1 of

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4 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

Figure 1. The `p-bubbles with p = 3 (left) and p = 100 (right). In

blue we outlined three horizontal lifts of `p-circles foliating the `p-

bubble.

class C1 (equivalently, of its boundary ∂E) is

C (E)= C (∂E) = {p ∈ ∂E : Tp∂E = D(p)}. (1.6)

In Section 7 we characterize the structure of C (E) for a C2-smooth φ-isoperimetric

set E ⊂ H1, proving that C (E) is made of isolated points. For the more general case

of φ-critical surfaces we obtain the following result, that we prove by adapting to the

sub-Finsler case the theory of Jacobi fields of [27]. Any φ-critical surface has constant

φ-curvature and the definition is presented in Section 7.

Theorem 1.2. Let φ and φ∗ be of class C2 and let Σ ⊂ H1 be a complete and

oriented surface of class C2. If Σ is φ-critical with non-vanishing φ-curvature then

C (Σ) consists of isolated points and C2 curves that are either horizontal lines or

horizontal lifts of simple closed curves.

The simple closed curves of Theorem 1.2 are described by a suitable ordinary

differential equation. We expect that these curves are φ†-circles, where φ† is the norm

defined as

φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2.

Here and hereafter, ⊥: R2 → R2 denotes the perp-operator ⊥(ξ) = ξ⊥, with

ξ⊥ = (x, y)⊥ = (−y, x), ξ = (x, y) ∈ R2.

Theorem 1.1 then follows by combining the results of Sections 4.2, 5, and 7. In

particular, starting from a first variation analysis, we establish a foliation property

outside the characteristic set for C2-smooth φ-isoperimetric sets (and more generally

for constant φ-curvature surfaces). Theorem 1.2 is a key step for concluding the proof.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 5

We also identify an explicit relation between φ-isoperimetric sets and geodesics in

the ambient space. In Section 6, we show that, outside the characteristic set, φ-

isoperimetric sets are foliated by sub-Finsler geodesics in H1 relative to the norm φ†.

We refer to Corollary 6.4 for a statement of the result. Notice that when φ = | · | is the

Euclidean norm, φ† reduces to | · |, and we recover the foliation by sub-Riemannian

geodesics of C2-smooth | · |-isoperimetric sets.

The regularity of the candidate isoperimetric sets Eφ is a major issue that we treat

in Section 8. While it is rather easy to check that φ-Pansu bubbles have the same

regularity as φ outside the characteristic set (at least if φ-circles are strictly convex,

see Lemma 8.1), it is not clear what regularity is inherited from φ at characteristic

points. In Section 8.3 we prove the following.

Theorem 1.3. Assume that φ is of class C4 and that φ-circles have strictly positive

curvature. Then ∂Eφ is an embedded surface of class C2.

In the case where φ or φ∗ are not differentiable, Theorems 1.1 and 1.3 cannot be

applied in a direct way. In Corollary 5.5 we show that the foliation property by

horizontal lifts of φ-circles outside the characteristic set can be recovered when φ∗

is only piecewise C2, thus allowing to cover the case φ = `p for p > 2. For general

non-differentiable norms, our next result is conditioned to the validity of the following

conjecture.

Conjecture 1.4. For any norm φ of class C∞+ , φ-isoperimetric sets are of class C2.

Here, a norm φ in R2 is said to be of class C∞+ if φ ∈ C∞(R2\{0}) and φ-circles have

strictly positive curvature. The proof of the following result is presented in Section 9.

Theorem 1.5. Assume that Conjecture 1.4 holds true. Then for any norm φ in R2

the φ-bubble Eφ ⊂ H1 is φ-isoperimetric.

Of a particular interest is the case of a crystalline norm. A norm φ : R2 → [0,∞) is

called crystalline if the φ-circle Cφ = Cφ(0, 1) is a convex polygon centrally symmetric

with respect to the origin. Let v1, . . . , v2N ∈ Cφ be the ordered vertices of this polygon,

and denote by ei = vi − vi−1, i = 1, . . . , 2N , the edges of Cφ, where v0 = v2N . We

consider the left-invariant vector fields

Xi := ei,1X + ei,2Y, i = 1, . . . , 2N, (1.7)

where ei = (ei,1, ei,2), and we notice that Xi+N = −Xi for i = 1, . . . , N . By a first

variation argument, we deduce a foliation property for φ-isoperimetric sets by integral

curves of the Xi, see Section 4.3.

Theorem 1.6. Let E ⊂ H1 be φ-isoperimetric for a crystalline norm φ. Let A ⊂ H1

be an open set such that ∂E ∩A is a connected z-graph of class C2. Then there exists

i = 1, . . . , N such that ∂E ∩ A is foliated by integral curves of Xi.

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6 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Geodesics of sub-Finsler structures on the Heisenberg group and other Carnot

groups have been studied in several papers (see, in particular, [2, 5, 6, 20, 29]).

Unfortunately, Theorem 1.6 does not provide enough information in order to establish

the global foliation property by φ†-geodesics in the crystalline case.

1.1. Structure of the paper. In Section 2 we introduce the notion of sub-Finsler

perimeter and we deduce a representation formula for Lipschitz sets (see Proposi-

tion 2.2), holding for any norm φ in R2. In Section 3 we prove existence of φ-

isoperimetric sets for a general norm φ, following the arguments in [19]. In Section 4

we derive first-variation necessary conditions for φ-isoperimetric sets, both when φ∗

is of class C1 (see Section 4.2) and when φ∗ is not differentiable (see Section 4.3).

In the former case, we introduce the notion of φ-curvature of a C2-smooth surfaces

(when φ∗ is C2) and of φ-critical surface. In the latter case, we deduce the (partial)

foliation property stated in Theorem 1.6 for crystalline norms. In Section 5 we deduce

a foliation property outside the characteristic set for φ-isoperimetric sets of class C2,

assuming φ and φ∗ to be regular enough. We then study such a foliation from the

point of view of geodesics in the ambient space in Section 6, and in Section 7 we study

the characteristic set of C2-smooth φ-critical surfaces and of φ-isoperimetric sets, as-

suming φ and φ∗ to be C2 (Theorem 1.2). In Section 8 we then prove Theorem 1.1

and we study regularity of φ-bubbles, summarized in Theorem 1.3. Finally, Section 9

is dedicated to general norms and contains the proof of Theorem 1.5.

Acknowledgments. The authors thank M. Ritore and C. Rosales for pointing out

a gap in a preliminary version of the paper. The first and third authors acknowl-

edge the support of ANR-15-CE40-0018 project SRGI - Sub-Riemannian Geometry

and Interactions. The first author acknowledges the support received from the Eu-

ropean Union’s Horizon 2020 research and innovation programme under the Marie

Sklodowska-Curie grant agreement No. 794592, of the INdAM–GNAMPA project

Problemi isoperimetrici con anisotropie, and of a public grant of the French National

Research Agency (ANR) as part of the Investissement d’avenir program, through the

iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.

2. Sub-Finsler perimeter

In this section, we introduce the notion of φ-perimeter in H1 for a norm φ in R2.

We start by fixing the notation relative to horizontal vector fields and sub-Finsler

norms in H1.

A smooth horizontal vector field is a vector field V on R3 that can be written as

V = aX + bY where a, b ∈ C∞(H1). When A ⊂ H1 is an open set and a, b ∈ C∞c (A)

have compact support in A we shall write V ∈ Dc(A). We fix on D(H1) the scalar

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 7

product 〈·, ·〉D that makes X, Y pointwise orthonormal. Then each fiber D(p) can be

identified with the Euclidean plane R2.

Let φ : R2 → [0,∞) be a norm. We fix on D(H1) the left-invariant norm associated

with φ. Namely, with a slight abuse of notation, for any p ∈ H1 and with a, b ∈ R we

define

φ(aX(p) + bY (p)) = φ((a, b)

).

Since the Haar measure of H1 is the Lebesgue measure of R3, the divergence in H1 is

the standard divergence. Therefore, for a smooth horizontal vector field V = aX+bY

we have div(V ) = Xa+ Y b.

Definition 2.1. The φ-perimeter of a Lebesgue measurable set E ⊂ H1 in an open

set A ⊂ H1 is

Pφ(E;A) = sup

{∫E

div(V )dp : V ∈ Dc(A) with maxξ∈A

φ(V (ξ)) ≤ 1

}.

When Pφ(E;A) <∞ we say that E has finite perimeter in A. When A = H1, we let

Pφ(E) = Pφ(E;H1).

Since all the left-invariant norms in the horizontal distribution are equivalent, we

have Pφ(E) <∞ if and only if the set E has finite horizontal perimeter in the sense

of [7, 16, 17].

For regular sets, we can represent Pφ(E) integrating on ∂E a kernel related to the

normal. Let νE be the Euclidean unit inner normal to ∂E. We define the horizontal

vector field NE : ∂E → D(H1) by

NE = 〈νE, X〉X + 〈νE, Y 〉Y,

where 〈·, ·〉 denotes the Euclidean scalar product in R3.

Proposition 2.2 (Representation formula). Let E ⊂ H1 be a set with Lipschitz

boundary. Then for every open set A ⊂ H1 we have

Pφ(E;A) =

∫∂E∩A

φ∗(NE) dH 2, (2.1)

where H 2 is the standard 2-Hausdorff measure in R3.

Proof. Let V ∈ Dc(A) be such that φ(V ) ≤ 1. By the standard divergence theorem

and by the definition of dual norm, we have∫E

div(V ) dξ = −∫∂E

〈V,NE〉D dH 2 ≤∫∂E∩A

φ(V )φ∗(NE)dH 2

≤∫∂E∩A

φ∗(NE)dH 2.

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8 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

By taking the supremum over all admissible V we then obtain

Pφ(E;A) ≤∫∂E∩A

φ∗(NE)dH 2.

To get the opposite inequality it is sufficient to prove that for every ε > 0 there

exists V ∈ Dc(A) such that φ(V ) ≤ 1 and

−∫∂E

〈V,NE〉D dH 2 ≥∫∂E∩A

φ∗(NE)dH 2 − ε.

Here, without loss of generality, we assume that A is bounded. We will construct

such a V with continuous coefficients and with compact support in A. The smooth

case V ∈ Dc(A) will follow by a standard regularization argument.

Let us define the sets

U ={p ∈ ∂E ∩ A : NE(p) is defined

}, Z =

{p ∈ U : NE(p) = 0

}.

From the results of [4] it follows that Z has vanishing H 2-measure. For any p ∈U \Z we take V ∈ D(p) such that φ(V ) = 1 and

〈V,NE〉D = φ∗(NE).

In general, this choice is not unique. However, there is a selection p 7→ V (p) that

is measurable (this follows since the coordinates are measurable, see for instance [3,

Theorem 8.1.3]). We extend V to Z letting V = 0 here. This extension is still

measurable.

Since ∂E ∩ A has finite H 2-measure, by Lusin’s theorem there exists a compact

set Kε ⊂ ∂E ∩A such that H 2((∂E ∩A) \Kε

)< ε and the restriction of V to Kε is

continuous. Now, by Tietze–Uryshon theorem we extend V from Kε to A in such a

way that the extended map, still denoted by V , is continuous with compact support

in A and satisfies φ(V ) ≤ 1 everywhere.

Our construction yields the following∫∂E∩A

φ∗(NE)dH 2 =

∫Kε

〈V,NE〉D dH 2 +

∫(∂E∩A)\Kε

φ∗(NE)dH 2

=

∫∂E∩A

〈V,NE〉D dH 2 −∫

(∂E∩A)\Kε

(〈V,NE〉D − φ∗(NE)) dH 2

≤∫∂E∩A

〈V,NE〉D dH 2 + Cε.

In the last inequality we used the fact that 〈V,NE〉D − φ∗(NE) is bounded and

H 2((∂E ∩ A) \Kε

)< ε. The claim follows. �

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 9

3. Existence of isoperimetric sets

For a measurable set E ⊂ H1 with positive and finite measure and a given norm φ

on R2 we define the φ-isoperimetric quotient as

Isopφ(E) =Pφ(E)

L 3(E)34

,

where L 3 denotes the Lebesgue measure of R3.

The isoperimetric quotient is invariant under left-translation (w.r.t. the operation

in (1.1)), i.e., Isopφ(p ∗E) = Isopφ(E) for any p ∈ H1 and E ⊂ H1 admissible, and it

is 0-homogeneous with respect to the one-parameter family of automorphisms (1.4),

i.e., Isopφ(λE) = Isopφ(E), where λE = δλ(E).

The isoperimetric problem (1.3) is then equivalent to minimizing the isoperimetric

quotient among all admissible sets. Namely, given m ∈ (0,∞), any isoperimetric set

E ⊂ H1 with L 3(E) = m is a solution to

CI = inf{

Isopφ(E) : E ⊂ H1 measurable, 0 < L 3(E) <∞}, (3.1)

and, vice versa, any solution E ⊂ H1 to (3.1) solves (1.3) within its volume class, i.e.,

with m = L 3(E). In particular, we have

CI = inf{Pφ(E) : E ⊂ H1 measurable, L 3(E) = 1

}. (3.2)

The constant CI depends on φ.

Since Pφ is equivalent to the standard horizontal perimeter, the isoperimetric in-

equality in [17] implies that CI > 0 and the validity of the following inequality for

any measurable set E with finite measure:

Pφ(E) ≥ CIL3(E)

34 . (3.3)

The constant CI is the largest one making true the above inequality and isoperimetric

sets are precisely those for which (3.3) is an equality.

Theorem 3.1 (Existence of isoperimetric sets). Let φ be any norm on R2. There

exists a set E ⊂ H1 with non-zero and finite φ-perimeter such that

Pφ(E) = CIL3(E)

34 . (3.4)

Theorem 3.1 follows by applying the strategy of [19, Section 4]. In the sequel we

denote the left-invariant homogeneous ball centered at p ∈ H1 of radius r > 0 by

B(p, r).

Proof of Theorem 3.1. We give a sketch of the proof. By perimeter and volume ho-

mogeneity with respect to {δλ}λ∈R it is enough to prove the existence of a minimizing

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10 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

set in the class of volume L 3(E) = 1. Let {Ek}k∈N be a minimizing sequence for

(3.2) such that for k ∈ N we have

L 3(Ek) = 1, Pφ(Ek) ≤ CI

(1 +

1

k

).

Assume that there exists m0 ∈ (0, 1/2) such that for any k ∈ N there exists pk ∈ H1

satisfying

L 3(Ek ∩B(pk, 1)) ≥ m0. (3.5)

Then, the translated sequence {−pk∗Ek}k∈N, still denoted {Ek}k∈N, is also minimizing

for (3.2) and satisfies L 3(Ek ∩B(0, 1)) ≥ m0.

Since Pφ is equivalent to the standar horizontal perimeter, we have a compactness

theorem for sets of finite φ-perimeter as in [17, Theorem 1.28]. Then, we can extract a

sub-sequence, still denoted {Ek}k∈N, converging in the L1loc(H1) sense to a set E ⊂ H1

of finite φ-perimeter. The lower semi-continuity of Pφ therefore implies

Pφ(E) ≤ lim infk→∞

Pφ(Ek) ≤ CI .

Moreover, we have

L 3(E) ≤ lim infk→∞

L 3(Ek) = 1 and

L 3(E ∩B(0, 1)) = limk→∞

L 3(Ek ∩B(0, 1)) ≥ m0.(3.6)

To prove (3.4) we are left to show that L 3(E) = 1, which follows by applying a

sub-Finsler version of [19, Lemma 4.2], ensuring existence of a radius R > 0 such that

L 3(E ∩ B(0, R)) = 1. This is based on (3.6) and on a canonical relation between

perimeter and derivative of volume in balls with respect to the radius, which is valid

in quite general metric structures, including sub-Finsler ones, see [1, Lemma 3.5].

We conclude by justifying the assumption (3.5). This follows by a sub-Finsler

version of [19, Lemma 4.1]. Using once more the equivalence of Pφ with the standard

horizontal perimeter, we deduce from [17, Theorem 1.18] the validity of the following

relative isoperimetric inequality holding for a constant C > 0 and any measurable set

E with finite measure

min{

L 3(B ∩ E)34 ,L 3(B \ E)

34

}≤ CPφ(E, λB),

where λ ≥ 1 is a constant depending only on φ, and B is any left-invariant homoge-

neous ball. Together with the fact that the family {B(p, λ) : p ∈ H1} has bounded

overlap, we can reproduce the argument of [19, Lemma 4.1] and prove the claim. �

Remark 3.2. Following the arguments of [19, Lemma 4.2], one also shows that any

isoperimetric set is equivalent to a bounded and connected one (i.e., it is bounded

and connected up to sets of zero Lebesgue measure).

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 11

4. First variation of the isoperimetric quotient

In this section we derive a first order necessary condition for φ-isoperimetric sets,

both when φ is regular or not.

4.1. Notation. We now introduce some notation that will be used throughout the

paper.

Let E,A ⊂ H1 be sets such that E is closed, A is open and there exists a function

g ∈ C1(A), called defining function for ∂E∩A, such that ∂E∩A = {p ∈ A : g(p) = 0}and ∇g(p) 6= 0 for every p ∈ ∂E ∩A. We say that E ∩A is a z-subgraph if there exist

an open set D ⊂ R2 and a function f ∈ C1(D), called graph function for ∂E ∩ A,

such that

E ∩ A = {(ξ, z) ∈ A : ξ ∈ D and z ≤ f(ξ)}.

In this case, g(ξ, z) = f(ξ)− z is a defining function for ∂E ∩ A.

The definition of z-epigraph is analogous and all results given below for z-subgraphs

have a straightforward counterpart for z-epigraphs. In a similar way, one can also

define x-subgraphs, y-subgraphs, x-epigraphs, and y-epigraphs.

Given a function g ∈ C1(A), we denote by G = (Xg)X + (Y g)Y the horizontal

gradient of g and we define the projected horizontal gradient as

G = (Xg, Y g) ∈ R2. (4.1)

If ∂E ∩ A is a z-graph with graph function f ∈ C1(D), we define F : D → R2 by

F (ξ) = G(ξ, f(ξ)) = ∇f(ξ)− 1

2ξ⊥, (4.2)

and

C (f) = {ξ ∈ D : F (ξ) = 0}. (4.3)

Hence C (E) ∩A = {(ξ, f(ξ)) : ξ ∈ C (f)}, where C (E) is the characteristic set of E,

defined in (1.6). The set C (f) has zero Lebesgue measure in D.

If E ∩A is the z-subgraph of a function f ∈ C1(D), by the representation formula

(2.1) we have

Pφ(E;A) =

∫D

φ∗(F (ξ))dξ.

When the dual norm φ∗ is of class C1, starting from a graph function f ∈ C1(D)

we define the vector field Xf : D → R2 by

Xf(ξ) = ∇φ∗(F (ξ)), ξ ∈ D.

The geometric meaning of the vector field Xf will be clarified in the next section,

see (5.2).

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12 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Remark 4.1. At any point ξ ∈ D such that F (ξ) 6= 0 the vector field Xf satisfies

φ(Xf(ξ)) = 1, (4.4)

since the gradient of φ∗ at any nonzero point has norm φ equal to one (even when φ∗

is not regular, by replacing the gradient by any element of the subgradient; see, for

instance, [18, Example 3.6.5]).

4.2. Regular norms.

Proposition 4.2 (First variation for isoperimetric sets). Let φ be a norm such that

φ∗ is of class C1. Let E ⊂ H1 be a φ-isoperimetric set such that, for some open set

A ⊂ H1, E ∩ A is a z-subgraph of class C1. Then the graph function f ∈ C1(D)

satisfies in the weak sense the partial differential equation

div(Xf

)= −3

4

Pφ(E)

L 3(E)in D. (4.5)

Proof. For small ε ∈ R and ϕ ∈ C∞c (D) let Eε ⊂ H1 be the set such that

Eε ∩ A = {(ξ, z) ∈ A : z ≤ f(ξ) + εϕ(ξ), ξ ∈ D},

and Eε \ A = E \ A. Starting from the representation formula

Pφ(Eε;A) =

∫∂Eε∩A

φ∗(NEε)dH2 =

∫D

φ∗(F + ε(Xϕ, Y ϕ))dξ, (4.6)

we compute the derivative

Pφ′ =

d

dεPφ(Eε;A)

∣∣∣∣ε=0

=

∫D

〈Xf, (Xϕ, Y ϕ)〉dξ =

∫D

〈Xf,∇ϕ〉dξ. (4.7)

On the other hand, the derivative of the volume is

V ′ =d

dεL 3(Eε)

∣∣∣∣ε=0

=

∫D

ϕdξ.

Inserting these formulas into

0 =d

Pφ(Eε)4

L 3(Eε)3

∣∣∣∣ε=0

=Pφ(E)3

L 3(E)4

(4Pφ

′L 3(E)− 3V ′Pφ(E)),

we obtain ∫D

〈Xf,∇ϕ〉dξ =3

4

Pφ(E)

L 3(E)

∫D

ϕdξ

for any test function ϕ ∈ C∞c (D). This is our claim. �

Proposition 4.2 still holds if we only have f ∈ Lip(D). If φ∗ is of class C2 and

f ∈ C2(D) then we have Xf ∈ C1(D \ C (f);R2). So equation (4.5) is satisfied

pointwise in D \ C (f) in the strong sense.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 13

Definition 4.3. Let f ∈ C2(D). We call the function Hφ : D \ C (f)→ R

Hφ(ξ) = div(Xf(ξ)

), ξ ∈ D \ C (f), (4.8)

the φ-curvature of the graph gr(f). We say that gr(f) has constant φ-curvature if

there exists h ∈ R such that Hφ = h on D \ C (f). Finally, we say that gr(f) is

φ-critical if there exists h ∈ R such that∫D

〈Xf,∇ϕ〉dξ = −h∫D

ϕdξ (4.9)

is satisfied for every ϕ ∈ C∞c (D).

Proposition 4.2 then asserts that the part of the boundary of a φ-isoperimetric set

of class C2 that can be represented as a z-graph is φ-critical and in particular it has

constant φ-curvature at noncharacteristic points.

Remark 4.4. Let us discuss how the proof of Proposition 4.2 can be adapted to the

case where E ∩ A is a x-subgraph of class C2. The case of y-subgraphs is analogous.

We have a defining function for ∂E ∩ A of the type g(x, y, z) = f(y, z) − x with

f ∈ C2(D). The projected horizontal gradient in (4.1) reads

G(y, z) =(− 1− 1

2yfz, fy +

1

2ffz

).

For ε ∈ R and ϕ ∈ C∞c (D) let Eε be the x-subgraph in A of f + εϕ. Then the

derivative of the φ-perimeter of Eε is

d

dεPφ(Eε;A)

∣∣∣∣ε=0

=

∫D

⟨∇φ∗(G),

(− yϕz/2, ϕy + (ϕf)z/2

)⟩dydz

= −∫D

ϕ(y, z) L f(y, z) dydz,

where L : C2(D)→ C(D) is the partial differential operator

L f =( ∂∂y

+f

2

∂z

)φ∗b(G)− y

2

∂zφ∗a(G), (4.10)

with ∇φ∗ = (φ∗a, φ∗b).

The statement for x-graphs is then that if E ⊂ H1 is φ-isoperimetric and E ∩A is

a x-subgraph with graph function f ∈ C2(D), then

L f = −3

4

Pφ(E)

L 3(E)in D.

When we only have f ∈ Lip(D), L f is well-defined in the distributional sense.

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14 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

4.3. Crystalline norms. In this section we focus on a norm φ having non-differentia-

bility points, and in particular on the case where it is crystalline. Recall that the dual

norm φ∗ to a non-differentiable one is not strictly convex, so that ∇φ∗ is constant on

subsets of R2 having nonempty interior.

Lemma 4.5. Let O be a subset of R2 where ∇φ∗ exists and is constant. Let E ⊂ H1

be such that E ∩ A = {(ξ, z) ∈ A : z ≤ f(ξ), ξ ∈ D} for some open set A ⊂ H1 and

f ∈ Lip(D). If F (ξ) ∈ O for almost every ξ ∈ D then E is not φ-isoperimetric.

Proof. As in the proof of Proposition 4.2, consider ϕ ∈ C∞c (D) and, for ε ∈ R small,

let Eε ⊂ H1 be the set such that

Eε ∩ A = {(ξ, z) ∈ A : z ≤ f(ξ) + εϕ(ξ), ξ ∈ D},

and Eε \ A = E \ A. Then, as in (4.7),

Pφ′ =

d

dεPφ(Eε;A)

∣∣∣∣ε=0

=

∫D

〈∇φ∗(F ),∇ϕ〉dξ.

By hypothesis, ∇φ∗(F ) is constant on D, so that Pφ′ = 0.

Now, choosing ϕ 6= 0 with constant sign, we deduce that

d

Pφ(Eε)4

L 3(Eε)3

∣∣∣∣ε=0

= −3Pφ(E)4

L 3(E)4

∫D

ϕ(ξ)dξ 6= 0,

contradicting the extremality of E for the isoperimetric quotient. �

We are ready for the proof of Theorem 1.6. This theorem disproofs Conjecture 8.0.1

in [30], where Pansu’s bubble was conjectured to solve the isoperimetric problem for

crystalline norms.

Let φ be a crystalline norm and denote by v1, . . . , v2N ∈ R2 the ordered vertices of

the polygon Cφ = Cφ(0, 1). Notice that vi+N = −vi for i = 1, . . . , N . The dual norm

φ∗ is also crystalline and the vertices of Cφ∗(0, 1) are in one-to-one correspondence

with the edges ei = vi − vi−1 of Cφ(0, 1) (with v0 = v2N). Namely, Cφ∗(0, 1) is the

convex hull of v∗1, . . . , v∗2N where, for i = 1, . . . , 2N , the vertex v∗i is the unique vector

of R2 such that

〈v∗i , ei〉 = 0 (4.11)

and 〈v∗i , vi〉 = 〈v∗i , vi−1〉 = 1. In particular, v∗i+N = −v∗i for i = 1, . . . , N .

Along the lines Li = Rv∗i , the norm φ∗ is not differentiable. In the positive convex

cone bounded by R+v∗i and R+v∗i+1 the gradient ∇φ∗ exists and is constant, and we

have ∇φ∗ = vi. For piecewise C1-smooth φ-isoperimetric sets the projected horizontal

gradient F takes values in L1 ∪ . . . ∪ LN , by Lemma 4.5.

Proof of Theorem 1.6. Let f ∈ C2(D) be the graph function of ∂E ∩ A. For i =

1, . . . , N , we let

Di = {ξ ∈ D : F (ξ) ∈ Li = Rv∗i }.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 15

If ξ ∈ Di then by (4.11) we have

F (ξ)⊥ ∈ R(v∗i )⊥ = Rei.

This implies that the vector field Xi in (1.7) is tangent to ∂E∩A at the point (ξ, f(ξ)).

We are going to prove the theorem by showing that D = Di for some i ∈ {1, . . . , N}.Notice that, for i, j ∈ {1, . . . , N} and i 6= j, vi and vj are linearly independent. By

Lemma 4.5 we have that D = ∪Ni=1Di. We claim, moreover, that

D = ∪Ni=1intDi. (4.12)

In order to check the claim, pick ξ ∈ D and assume by contradiction that ξ 6∈ intDi

for i = 1, . . . , N . Let i1 be such that ξ ∈ Di1 . Since ξ 6∈ intDi1 , for every ε > 0 the

set D\Di1 intersects the disc of radius ε centered at ξ. Hence, there exist i2 6= i1, and

a sequence (ξn)n∈N in Di2 \Di1 converging to ξ. Now, either ξn ∈ intDi2 for infinitely

many n or ξn 6∈ intDi2 for n large enough. In the first case ξ ∈ intDi2 , leading

to a contradiction. In the second case, we repeat the reasoning leading to (ξn)n∈N,

replacing Di1 by Di2 and ξ by ξn for every n ∈ N, and, by a diagonal argument,

we obtain i3 6= i1, i2, and a sequence (ξn)n∈N in Di3 \ (Di1 ∪ Di2) converging to ξ.

Repeating the argument finitely many times, we end up with iN ∈ {1, . . . , N} and

a sequence (ξn)n∈N in DiN \ (∪N−1j=1 Dij) converging to ξ with D = Di1 ∪ · · · ∪ DiN .

Since DiN \ (∪N−1j=1 Dij) = D \ (∪N−1

j=1 Dij) is open, we deduce that ξ ∈ intDiN . This

concludes the contradiction argument, proving (4.12).

Let vi and vj be linearly independent. We claim that

int(Di) ∩ int(Dj) = ∅. (4.13)

Consider the vector field X ′ on D×R defined by X ′(ξ, z) = (ei, ei,1fx(ξ)+ei,2fy(ξ)).

Then X ′ is C1 and both X ′ and Xj are tangent to ∂E ∩A in a neighborhood of any

point of Sj, where

Sk = {(ξ, f(ξ)) : ξ ∈ int(Dk)}, k = 1, . . . , N.

Hence [X ′, Xj] ∈ Tξ(∂E∩A) for every ξ ∈ Sj. On the other hand, X ′ coincides with Xi

on Si×R, and therefore [X ′, Xj] = [Xi, Xj] = cijZ on Si, with cij ∈ R \ {0}. Assume

by contradiction that Si ∩Sj contains at least one point ξ. By continuity of [X ′, Xj],

we deduce from the above reasoning that Z(ξ) ∈ Tξ(∂E ∩ A). The contradiction

comes from the remark that, by definition of Si and Sj, also Xi(ξ) and Xj(ξ) are in

Tξ(∂E ∩ A). We proved (4.13).

We deduce from (4.12) and (4.13) that {int(D1), . . . , int(DN)} is an open disjoint

cover of D. We conclude by connectedness of D. �

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16 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

5. Integration of the curvature equation

Throughout this section φ∗ is a norm of class C2, unless explicitly mentioned other-

wise.

Let A ⊂ H1 be open and g ∈ C1,1(A) be such that ∇g(p) 6= 0 for every p in

Σ = {p ∈ A : g(p) = 0}. The projected horizontal gradient G : A → R2 introduced

in (4.1) is Lipschitz continuous. Assume that Σ has no characteristic points, that is,

G(p) 6= 0 for every p ∈ Σ. We use the coordinates G = (a, b) with a, b ∈ Lip(A) and

we consider G⊥ = (−b, a). The horizontal vector field G ⊥ = −bX + aY is tangent to

Σ.

Definition 5.1. A curve γ ∈ C1(I; Σ) is said to be a Legendre curve of Σ if γ(t) =

G ⊥(γ(t)) for all t ∈ I.

In coordinates, a curve γ = (ξ, z) in Σ is a Legendre curve if and only if

ξ = G⊥(γ) and z = ω(ξ, ξ).

Since G ⊥ is Lipschitz continuous, the graph Σ is foliated by Legendre curves: for any

p ∈ Σ there exists a unique (maximal) Legendre curve passing through p.

Consider now the case where Σ is a z-graph with graph function f ∈ C1,1(D),

where D is an open subset of R2. Then G(ξ, f(ξ)) = F (ξ), where F is defined as in

(4.2), and a Legendre curve γ = (ξ, z) satisfies

ξ = F⊥(ξ) and z = ω(ξ, ξ). (5.1)

The domain D is foliated by integral curves of F⊥. On D we define the vector field

N ∈ Lip(D;R2) by

N (ξ) = Xf(ξ) = ∇φ∗(F (ξ)), ξ ∈ D. (5.2)

We know that φ(N ) = 1, by (4.4). We may call N the φ-normal to the foliation of

D by integral curves of F⊥. We denote by Hφ = div(N ) the divergence of N .

Theorem 5.2. Let φ∗ be of class C2. Let Σ be the z-graph of a function f ∈ C2(D)

with C (f) = ∅. Then any Legendre curve γ ∈ C1(I; Σ), with γ = (ξ, z), satisfies

d

dtN (ξ) = Hφ(ξ)ξ and z = ω(ξ, ξ). (5.3)

Proof. The second equality in (5.3) is part of the definition of a Legendre curve. We

prove the first equality.

We identify N (ξ) and ξ = F⊥(ξ) with column vectors and we denote by Jg

the Jacobian matrix of a differentiable mapping g. By the chain rule, using the

coordinates F = (a, b) and ξ = (−b(ξ), a(ξ)) we obtain

d

dtN (ξ) = H φ∗(F (ξ))JF (ξ)ξ =

(−baxφ∗aa − bbxφ∗ab + aayφ

∗aa + abyφ

∗ab

−baxφ∗ab − bbxφ∗bb + aayφ∗ab + abyφ

∗bb

), (5.4)

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 17

where H φ∗ is the Hessian matrix of φ∗ and the second order derivatives of φ∗ are

evaluated at F (ξ). Since φ∗ is of class C2, we identified φ∗ab = φ∗ba. By Euler’s

homogeneous function theorem, since ∇φ∗ is 0-positively homogeneous there holds

〈∇φ∗a(F ), F 〉 = 0 and 〈∇φ∗b(F ), F 〉 = 0. These formulas read

aφ∗aa + bφ∗ab = 0 and aφ∗ab + bφ∗bb = 0.

Plugging these relations into (5.4), we obtain

d

dtN (ξ) = (axφ

∗aa + bxφ

∗ab + ayφ

∗ab + byφ

∗bb) ξ. (5.5)

On the other hand, we have

div(N ) = div(Xf) = axφ∗aa + bxφ

∗ab + ayφ

∗ab + byφ

∗bb,

so that (5.5) yields the claim. �

Remark 5.3. An analogue of Theorem 5.2 holds true for x-graphs. Let Σ be a x-

graph Σ without characteristic points and with defining function g(x, y, z) = f(y, z)−x for some f of class C2. Let γ ∈ C1(I; Σ) be a Legendre curve with coordinates

γ(t) = (f(ζ(t)), ζ(t)) for t ∈ I and consider the vector N (y, z) = ∇φ∗(G(y, z)).

Following the same steps as in the proof of Theorem 5.2, one gets

d

dtN (ζ) = L f(ζ)G⊥(ζ) on I.

Hence, the conclusion of Theorem 5.2 holds with Hφ = div(Xf

)replaced by the

quantity L f defined in Remark 4.4. Notice that Hφ and L f coincide on surfaces

that are both x-graphs and z-graphs.

An analogous remark can be made for y-graphs.

Corollary 5.4. Let φ∗ be of class C2. Let Σ be the z-graph of a function f ∈ C2(D)

with C (f) = ∅. If Σ has constant φ-curvature h 6= 0 then it is foliated by Legendre

curves that are horizontal lift of φ-circles in D with radius 1/|h|, followed in clockwise

sense if h > 0 and in anti-clockwise sense if h < 0.

Proof. Having constant φ-curvature h means that

div(N ) = div(Xf) = h in D.

By Theorem 5.2, for any Legendre curve γ = (ξ, z) we have

d

dtN (ξ)−H(ξ)ξ = 0.

We may than integrate this equation and deduce that there exists ξ0 ∈ R2 such that

along ξ we have

N (ξ)− hξ = −hξ0. (5.6)

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18 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

From (4.4) and (5.2) we conclude that

|h|φ(ξ − ξ0) = φ(h(ξ − ξ0)) = φ(N ) = 1.

Finally, notice that 〈N (ξ), F (ξ)〉 > 0 if F (ξ) 6= 0, so that t 7→ F (ξ(t)) rotates

clockwise if h > 0 and anti-clockwise if h < 0, according to (5.3). Hence, t 7→ F (ξ(t))⊥

and t 7→ ξ(t) also rotate clockwise if h > 0, and anti-clockwise if h < 0. �

Let us discuss an extension of Corollary 5.4 to the case in which we replace the

assumption that φ∗ is C2 by the weaker assumption that φ∗ is piecewise C2, in the

following sense: there exist k ∈ N and A1, . . . , Ak ∈ R2 such that φ∗ is C2 on R2 \∪kj=1span(Aj).

A relevant case where this assumption holds true is when φ is the `p norm

`p(x, y) = (|x|p + |y|p)1p , x, y ∈ R,

with p > 2. Indeed, the dual norm (`p)∗ coincides with the norm `q, with q =

p/(p−1) < 2, which is C2 out of the coordinate axes, but not on the whole punctured

plane R2 \ {0}. We can prove the following.

Corollary 5.5. Let φ∗ be piecewise C2. Let Σ be the z-graph of a function f ∈ C2(D)

with C (f) = ∅. If Σ has constant φ-curvature h 6= 0 then it is foliated by Legendre

curves that are horizontal lifts of φ-circles in D with radius 1/|h|, followed in clockwise

sense if h > 0 and in anti-clockwise sense if h < 0.

Proof. Under the assumptions of the corollary, the projected horizontal gradient is

C1 on D and Legendre curves can be introduced as in Definition 5.1.

Consider any Legendre curve γ = (ξ, z) on Σ. Let us denote by I ⊂ R the maximal

interval of definition of γ and by J the open subset of I defined as follows: t ∈ J if and

only if F (ξ(t)) is in the region where φ∗ is C2. For the restriction of γ to a connected

component J0 of J , Theorem 5.2 can be recovered. In particular, since Σ has constant

φ-curvature h 6= 0, then γ|J0 is the lift of a φ-circle of radius 1/|h|, followed clockwise

or anti-clockwise depending on the sign of h. If t ∈ I \J , then F (ξ(t)) belongs to one

of the lines span(A1), . . . , span(Ak) on which φ∗ may lose the C2 regularity. Notice

that the restriction of ξ to a connected component of J compactly contained in I

follows an arc of φ-circle connecting two lines of the type span(Aj). In particular, it

cannot have arbitrarily small length.

If I \ J is made of isolated points, then γ : I → Σ is the lift of a φ-circle of radius

1/|h|. Indeed, an arc of φ-circle of prescribed radius followed in a prescribed sense is

only determined by its initial point and its tangent line there. Since γ is an arbitrary

Legendre curve on Σ, the proof is complete if show that I\J does not contain intervals

of positive length.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 19

Assume by contradiction that [t0, t1] is contained in J with t0 < t1. Then F (ξ(t))

is constantly equal to some A ∈ R2 for t ∈ [t0, t1]. Let δ > 0 and κ : (−δ, δ) → Σ

be a C1 curve such that κ(0) = γ(t0) and κ′(0) is not proportional to γ′(t0). Write

κ(s) = (ξs, zs) and notice that F (ξs) converges to A as s → 0. Consider for each

s ∈ (−δ, δ) the Legendre curve γs such that γs(t0) = κ(s). Then γs converges to γ

and F ◦ γs converges to F ◦ γ, uniformly on [t0, t1], as s→ 0. Hence, for ε > 0 and |s|small enough, the restriction of γs to (t0 + ε, t1− ε) cannot contain the lift of any arc

of φ-circle of radius 1/|h|. This implies that there exists a nonempty open region of

Σ of the form {γs(t) : t ∈ (t0 + ε, t1 − ε), |s| < δ} on which F (ξ) = A, contradicting

the assumption that Σ has constant nonzero φ-curvature. �

6. Foliation property with geodesics

In this section we prove that the Legendre foliation of a surface (a z-graph) with con-

stant φ-curvature consists of length minimizing curves in the ambient space (geodesics)

relative to the norm φ† in R2 defined by

φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2.

We consider a general norm ψ in R2 and for T ≥ 0 we introduce the class of curves

AT ={γ = (ξ, z) ∈ AC([0, T ];H1) : z = ω(ξ, ξ) and ψ(ξ) ≤ 1 a.e.

},

where ω is the symplectic form introduced in (1.2). In the sequel, we denote by

u = ξ ∈ L1([0, T ];R2) the control of γ. For given points p0, p1 ∈ H1 we consider the

optimal time problem

inf{T ≥ 0 : there exists γ ∈ AT such that γ(0) = p0 and γ(T ) = p1}. (6.1)

We call a curve γ realizing the minimum in (6.1) a ψ-time minimizer between p0 and

p1. In this case, we call the pair (γ, u) with u = ξ an optimal pair. A ψ-time minimizer

is always parameterized by ψ-arclength, i.e., ψ(u) = 1. So, ψ-time minimizers are

ψ-length minimizers parameterized by ψ-arclength.

An optimal pair (γ, u) satisfies the necessary conditions given by Pontryagin’s Max-

imum Principle. As observed in [6], it necessarily is a normal extremal, whose defi-

nition is recalled below. The Hamiltonian associated with the optimal time problem

(6.1) is H : H1 × R3 × R2 → R

H(p, λ, u) =(λx −

y

2λz

)u1 +

(λy +

x

2λz

)u2 = 〈λξ +

1

2λzξ

⊥, u〉,

where λ = (λξ, λz) ∈ R2 × R.

Definition 6.1. The pair (γ, u) ∈ AC([0, T ];H1)×L1([0, T ];R2) is a normal extremal

if there exists a nowhere vanishing curve λ ∈ AC([0, T ];R3) such that (γ, λ) solves

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20 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

a.e. the Hamiltonian system γ = Hλ(γ, λ, u)

λ = −Hp(γ, λ, u),

and for every t ∈ [0, T ] we have

1 = H(γ(t), λ(t), u(t)) = maxψ(u)≤1

H(γ(t), λ(t), u). (6.2)

In the coordinates γ = (ξ, z) and λ = (λξ, λz), the Hamiltonian system readsξ = u,

z = ω(ξ, u),

λξ = 12λzu

⊥,

λz = 0.(6.3)

Theorem 6.2. Let ψ be of class C1 and let γ = (ξ, z) ∈ AC([0, T ];H1) be a horizontal

curve. The following statements (i) and (ii) are equivalent:

(i) γ is a local ψ-length minimizer parametrized by ψ-arclength;

(ii) the pair (γ, u) with u = ξ is a normal extremal.

Moreover, if ψ is of class C2 then each of (i) and (ii) is equivalent to

(iii) γ is of class C2 and parameterized by ψ-arclength, and there is λ0 ∈ R such

that

H ψ(ξ)ξ = λ0ξ⊥, (6.4)

where H ψ is the Hessian matrix of ψ.

Proof. The equivalence between (i) and (ii) is [6, Theorem 1].

Let us show that (ii) implies (iii). We set

M (t) = λξ(t) +1

2λz(t)ξ(t)

⊥, t ∈ [0, T ], (6.5)

where λ = (λξ, λz) is the curve given by the definition of extremal. Then the maxi-

mality condition in (6.2) for normal extremals reads

1 = 〈M (t), u(t)〉 = maxψ(u)≤1

〈M (t), u〉 = ψ∗(M (t)). (6.6)

This is equivalent to the identity

M (t) = ∇ψ(u(t)). (6.7)

When ψ is of class C2, from (6.7), (6.5), and (6.3) we obtain the differential equation

for u = ξ

H ψ(u)u = ˙M = λξ +1

2λzξ +

1

2λzu

⊥ = λzu⊥. (6.8)

This is (6.4) with λ0 := λz.

Now we show that (ii) is implied by (iii). Consistently with (6.7), we define M (t) =

∇ψ(u(t)), for t ∈ [0, T ]. Then ψ∗(M ) = 1.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 21

We define the curve λ = (λξ, λz) letting λz = λ0 and λξ = M − 12λzξ

⊥. When ψ is

of class C2, we obtain

λξ = ˙M − 1

2λz ξ

⊥ = H ψ(ξ)ξ − 1

2λz ξ

⊥ =1

2λzu

⊥.

Hence, all equations in (6.3) are satisfied, showing that the pair (γ, u) is a normal

extremal. This proves that (iii) implies (ii). �

Remark 6.3. When λ0 6= 0, equation (6.4) can be integrated in the following way.

Using (6.8), the equation is equivalent to ˙M = λ0ξ⊥, that implies M = λ0(ξ⊥ − ξ⊥0 )

for some constant ξ0 ∈ R2. So from (6.6) we deduce that |λ0|ψ∗(ξ⊥ − ξ⊥0 ) = 1. If we

choose ψ = φ† then we have ψ∗(ξ⊥) = φ(ξ). So the previous equation becomes the

equation for a φ-circle

φ(ξ − ξ0) = 1/|λ0|.

Corollary 6.4. Let φ be a norm with dual norm φ∗ of class piecewise C2 and let

f ∈ C2(D) be such that C (f) = ∅. If gr(f) has constant φ-curvature, then it is

foliated by geodesics of H1 relative to the norm φ†.

The proof is Corollary 5.5, combined with Remark 6.3 and Theorem 6.2.

7. Characteristic set of φ-critical surfaces

In this section we study the characteristic set of φ-critical surfaces and then apply

the results to φ-isoperimetric sets. For a C2 surface Σ ⊂ H1, the characteristic set is

C (Σ) = {p ∈ ∂E : TpΣ = D(p)}. (7.1)

Note that any C2 surface Σ ⊂ H1 is a z-graph around any of its characteristic points

p ∈ C (Σ).

When Σ is oriented, the φ-curvature Hφ of Σ can be defined in a globally coherent

way. When Σ is a z-graph at the point p = (ξ, z) = (x, y, z) ∈ Σ we let Hφ(p) =

div(Xf)(ξ) where f is a z-graph function; when Σ is a x-graph, we let Hφ(p) =

L f(y, z), where now f is a x-graph function and L f is defined in (4.10); when Σ is

a y-graph we proceed analogously.

We say that Σ is φ-critical if it is closed, has constant φ-curvature and it is φ-critical

in a neighborhood of any characteristic point.

Our goal is to prove Theorem 1.2. The proof is obtained combining Lemma 7.1

and Theorem 7.2 below.

In this section, φ and φ∗ are two norms of class C2. We will omit to mention this

assumptions in the various statements.

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22 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

7.1. Qualitative structure of the characteristic set.

Lemma 7.1. Let Σ ⊂ H1 be a C2 surface with constant φ-curvature. Then C (Σ)

consists of isolated points and C1 curves. Moreover, for every isolated point p0 =

(ξ0, z0) ∈ C (Σ) and every f such that p0 ∈ gr(f) ⊂ Σ, we have rank(JF (ξ0)) = 2,

where F is the projected horizontal gradient introduced in (4.2).

Proof. We let C (f) be as in (4.3). For any ξ0 ∈ C (f), the Jacobian matrix JF (ξ0)

has rank 1 or 2. Indeed, an explicit calculation shows that JF (ξ0) 6= 0 for all ξ0 ∈ D.

If rank(JF (ξ0)) = 2 then ξ0 is an isolated point of C (f).

We study the case rank(JF (ξ0)) = 1. We claim that in this case C (f) is a curve of

class C1 in a neighborhood of ξ0. The argument that we use here is inspired by [8].

For b ∈ R2 we define Fb : D → R, Fb = 〈F, b〉. When b /∈ ker(JF (ξ0)), the equation

Fb = 0 defines a C1 curve Γb near and through ξ0. We have C (f) ⊂ Γb. Since ∇Fb(ξ0)

is in the image of JF (ξ0), which is a line independent of b, the normal direction to

Γb at ξ0 does not depend on b. We choose one of the two unit normals and we call it

N ∈ R2.

We claim that there exist a, b ∈ S1, where S1 = {w ∈ R2 : |w| = 1}, such that

a /∈ {b,−b}, a, b /∈ ker(JF (ξ0)), |〈∇φ∗(b⊥), N〉| 6= |〈∇φ∗(a⊥), N〉|. (7.2)

To prove the claim pick b ∈ S1 \ker(JF (ξ0)) (this is possible since rank(JF (ξ0)) 6= 0),

and define the set

Kb :={v ∈ Cφ : |〈v,N〉| = |〈∇φ∗(b⊥), N〉|

}.

Since the map ∇φ∗ : S1 → Cφ is continuous, the set (∇φ∗)−1(Kb) ⊂ S1 is closed in

S1. Moreover, ∇φ∗ : S1 → Cφ is surjective, since for every w ∈ Cφ and every v in

the subgradient of φ∗ at w, we have w = ∇φ∗(v) (see, e.g., [28, Theorem 23.5]). As

a consequence, (∇φ∗)−1(Kb) 6= S1, since otherwise we would have Kb = Cφ, which is

impossible. The set

Υ = ker(JF (ξ0))⊥ ∪ (∇φ∗)−1(Kb) ∪ {b⊥,−b⊥}

is therefore a proper closed subset of S1, and the claim follows by choosing a⊥ ∈ S1\Υ.

Fix a, b ∈ S1 such that (7.2) holds and, for α ∈ (0, 1), let Cα := {v ∈ R2 : |〈N, v〉| <|v| sinα} be the cone centered at ξ0 with axis parallel to N⊥ and aperture 2α. Since

Γa,Γb are C1, there exists δ ∈ (0, 1) such that

{ξ ∈ Γa ∪ Γb : |ξ − ξ0| < δ} ⊂ Cα,δ, (7.3)

where we set Cα,δ = {ξ ∈ Cα : |ξ − ξ0| < δ}.Let us assume by contradiction that C (f) is not a C1 curve near ξ0. Then there

exists a nonempty connected component A of Cα,δ \ (Γa ∪ Γb) such that, letting

Λa = Γa ∩ ∂A, Λb := Γb ∩ ∂A, Λ∂ := ∂{|ξ − ξ0| < δ} ∩ ∂A,

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 23

Figure 2. The cone Cα and the region A. On the left, A does not

touch ∂{|ξ−ξ0| < δ}, while it does on the right. We can always restrict

our attention to the case on the left when ξ0 is a density point of C (f).

Figure 3. Proportions in Cα,δ.

we have

Λa 6= ∅, Λb 6= ∅, ∂A = Λa ∪ Λb ∪ Λ∂, ](Λa ∩ Λb) ≤ 2. (7.4)

See Figure 2. Notice that A, Λa, Λb, and Λ∂ depend on δ. By (7.3) (see also Figure 3),

we have

L 2(A) ≤ δ2 tan(α). (7.5)

By (7.4) and since C (f) ⊂ Λa∩Λb, for ξ ∈ int(Λa)∪int(Λb) we have F (ξ) 6= 0, where

we endow Λa and Λb with their relative topologies. We deduce that F (ξ) = ca(ξ)a⊥

with ca(ξ) 6= 0 for ξ ∈ int(Λa) and F (ξ) = cb(ξ)b⊥ with cb(ξ) 6= 0 for ξ ∈ int(Λb).

Using the fact that ∇φ∗ is positively 0-homogeneous it then follows that the vector

field N : D \C (f)→ R2, N (ξ) = ∇φ∗(F (ξ)), is constant along Λa and Λb. Namely,

N (ξ) = sgn(ca)∇φ∗(a⊥) =: Na, ξ ∈ int(Λa),

N (ξ) = sgn(cb)∇φ∗(b⊥) =: Nb, ξ ∈ int(Λb).

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24 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

By assumption, and since φ∗ ∈ C2, there exists a constant h ∈ R such that

div(N (ξ)) = h, x ∈ D \ C (f),

in the strong sense. Then by the divergence theorem, and since A ∩ C (f) = ∅, we

have

hL 2(A) =

∫A

div(N )dxdy =

∫Λa

〈Na, Na〉dH 1+

∫Λb

〈Nb, Nb〉dH 1+

∫Λ∂

〈N , N∂〉dH 1,

where Na, Nb, and N∂ are, respectively, the normals to Λa, Λb, and Λ∂, exterior with

respect to A. For α→ 0+ we have∫Λa

NaH1 = δ(−N + o(1)),∫

Λb

NbH1 = δ(N + o(1)),∣∣∣∣∫

Λ∂

〈N , N∂〉dH 1

∣∣∣∣ ≤ Cδα,

where o(1) → 0 as α → 0+ and C > 0 denotes a suitable constant. Now from (7.5)

we deduce that

|δ tan(α)h| ≥ |〈Nb −Na, N〉+ o(1)| − Cα,that implies 〈Nb −Na, N〉 = 0 in contradiction with (7.2).

This proves that C (f) is a C1 curve around any point ξ0 with rank(JF (ξ0)) = 1. �

7.2. Characteristic curves in φ-critical surfaces. Given a surface Σ ⊂ H1, we

call a characteristic curve on Σ any (nontrivial) curve Γ ⊂ C (Σ). In this section we

prove the following result.

Theorem 7.2. Let Σ be a complete and oriented surface of class C2. If Σ is φ-critical

with non-vanishing φ-curvature h 6= 0 then any characteristic curve on Σ is either a

horizontal line or the horizontal lift of a simple closed curve.

For a characteristic curve Γ in Σ we denote its coordinates by Γ = (Ξ, ζ) ∈ R2×R.

For any p0 = (ξ0, z0) on Γ, let δ > 0 be small enough to have

{ξ ∈ R2 : |ξ − ξ0| < δ} \ supp(Ξ) = B+ ∪B−, (7.6)

where B+, B− ⊂ R2 are disjoint open connected sets. The φ-normal N in (5.2) is

well-defined in B+ ∪B−.

Lemma 7.3. Let Σ be a C2 surface with constant φ-curvature. With the above nota-

tion, the following limits exist

N ±(ξ0) := limB±3ξ→ξ0

N (ξ) (7.7)

and satisfy N +(ξ0) = −N −(ξ0).

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 25

Proof. This is a straightforward corollary of [8, Proposition 3.5]. �

Proposition 7.4. Let Σ be a φ-critical surface of class C2 and let Γ = (Ξ, ζ) be a

characteristic curve on Σ. Then for every p0 = (ξ0, z0) in Γ we have

N ±(ξ0) ∈ Tξ0Ξ, (7.8)

where N ± is defined as in Lemma 7.3.

Proof. Let f ∈ C2(D) be a graph function for Σ with ξ0 ∈ D ⊂ R2. Without loss of

generality we assume D = {|ξ − ξ0| < δ} and let D± := D ∩B±, where B± are as in

(7.6). Let h ∈ R be the φ-curvature of Σ. Since Σ is φ-critical, for any ϕ ∈ C∞c (D)

we have ∫D

〈Xf,∇ϕ〉 dξ = −∫D

hϕ dξ

and div(Xf) = h pointwise in D+ ∪ D−. Then, denoting by NΞ the normal to Ξ

pointing towards D−, by the divergence theorem we have∫D

hϕ dξ =

∫D+

div(Xf)ϕ dξ +

∫D−

div(Xf)ϕ dξ

= −∫D+∪D−

〈Xf,∇ϕ〉 dξ +

∫Ξ

ϕ〈N +, NΞ〉 dH 1 −∫

Ξ

ϕ〈N −, NΞ〉 dH 1

=

∫D

hϕ dξ +

∫Ξ

ϕ〈N + −N −, NΞ〉 dH 1.

By Lemma 7.3, this implies that∫Ξ

ϕ〈N +, NΞ〉 dH 1 = 0

and since ϕ is arbitrary, this yields the claim. �

Remark 7.5. Under the assumptions of the previous proposition, the characteristic

curves Γ = (Ξ, ζ) of ∂E are of class C2. This can be proved exactly as in Proposi-

tion 4.20 of [27] using condition (7.8). In particular, Ξ is of class C2.

7.2.1. Parametrization of constant φ-curvature surfaces around characteristic curves.

In this section, we study a φ-critical surface Σ of class C2 having constant φ-curvature

h 6= 0 near a characteristic curve. Without loss of generality we assume h > 0.

We assume φ to be normalized in such a way that φ(1, 0) = 1 and we fix a

parametrization µ : [0,M ]→ R2 of Cφ such that φ†(µ) = 1, µ([0,M ]) = Cφ, with ini-

tial and end-point µ(0) = µ(M). We choose the clockwise orientation and we extend

µ to the whole R by M -periodicity. We have µ ∈ C2(R;R2) and

µ(τ) = ∇φ∗(µ(τ)⊥), for all τ ∈ R. (7.9)

In fact, letting N (t) = ∇φ∗(µ(t)⊥), we have ˙N = µ as in (5.3). Equation (7.9) then

follows by integration using the fact that 0 is the center of Cφ.

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26 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Let Γ = (Ξ, ζ) ∈ C2(I; Σ) be a characteristic curve parameterized in such a way

that

φ(Ξ) = 1 on I. (7.10)

Locally, Γ disconnects Σ and there are no other characteristic points of Σ close to Γ,

by Lemma 7.1.

According to Corollary 5.4, Σ \ C (Σ) admits near Γ a Legendre foliation made of

horizontal lifts of φ-circles of radius 1/h, followed in the clockwise sense. Hence, given

a point (ξ0, z0) ∈ Σ \ C (Σ) near Γ, there exist c ∈ R2 and τ ∈ [0,M ] such that the

horizontal lift of

ξ(s) = c+ h−1µ(τ + hs)

passing through (ξ0, z0) at s = 0 stays in Σ until it meets a characteristic point.

Here, c is the center of the φ-circle. Notice that ∇φ∗(ξ(s)⊥) = N (ξ(s)), so that,

by Lemma 7.3 and (7.8), ∇φ∗(ξ(0)⊥) converges to a vector collinear to Ξ(t) as ξ0

approaches Ξ(t) for some t ∈ I. By (4.4) and (7.10), ∇φ∗(ξ(0)⊥) converges either to

Ξ(t) or to −Ξ(t) as ξ0 approaches Ξ(t). Since Ξ locally disconnects the plane, we can

fix a side from where ξ0 approaches Ξ and, up to reversing the parameterization of Γ,

we can assume that ∇φ∗(ξ(0)⊥) converges to Ξ(t) as ξ0 converges to Ξ(t). Thanks to

(7.9) and since ξ(0) = µ(τ), we deduce that µ(τ) = ∇φ∗(ξ(0)⊥) converges to Ξ(t) as

ξ0 → Ξ(t). In particular, the limit direction of ξ(0) as ξ0 → Ξ(t) is transversal to Ξ.

By local compactness of the set of φ-circles with radius 1/h, the horizontal lift

passing through Γ(t) at s = 0 of a curve c + h−1µ(τ + hs) with µ(τ) = Ξ(t) is a

Legendre curve contained in Σ, for s either in a positive or a negative neighborhood

of 0. To fix the notations, we assume that s is in a positive neighborhood of 0, the

computations being equivalent in the other case. Moreover, there is no other Legendre

curve having Γ(t) in its closure and whose projection on the xy-plane stays in the

chosen side of Ξ, since τ ∈ [0,M) and c ∈ R2 are uniquely determined by

µ(τ) = Ξ(t), c = Ξ(t)− h−1µ(τ) = Ξ(t)− h−1Ξ(t).

It is then possible to parameterize locally near Γ one of the two connected compo-

nents of Σ \ Γ by Legendre curves using the function

(t, s) 7→ γ(t, s) = (ξ(t, s), z(t, s)) (7.11)

where

ξ(t, s) = h−1µ(τ(t) + hs) + Ξ(t)− h−1Ξ(t), t ∈ I, s > 0, (7.12)

with τ uniquely defined via the equation

µ(τ(t)) = Ξ(t), t ∈ I, (7.13)

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 27

and z defined by

z(t, s) = ζ(t) +

∫ s

0

ω(ξ(t, σ), ξs(t, σ))dσ. (7.14)

As discussed above, we have

∇φ∗(ξs(t, 0)⊥) = Ξ(t), (7.15)

φ†(ξs) = 1. (7.16)

For t ∈ I, we define the characteristic time s(t) as the first positive time s > 0

such γ(t, s(t)) ∈ C (Σ). We will prove later that such a s(t) exists. Finally, we let

S := {(t, s) : t ∈ I, 0 ≤ s ≤ s(t)} and we consider the surface γ(S) ⊂ Σ.

Lemma 7.6. We have γ ∈ C1(S; Σ) with γ(·, 0) = Γ. Moreover, the second order

derivatives γss, γts, γst are well-defined and

γts = γst. (7.17)

Proof. By (7.12) and (7.14), we see that γss exists and that ξts = ξst. Moreover,

zst = ω(ξt(t, ·), ξs(t, ·)) + ω(ξ(t, ·), ξst(t, ·))

= ω(ξt(t, ·), ξs(t, ·)) + ω(ξ(t, ·), ξts(t, ·)) = zts.

On the surface γ(S) we consider the vector field

V (t, s) := γt(t, s) = (ξt(t, s), zt(t, s)) ∈ R3. (7.18)

It plays the role of the Jacobi vector field V in [27, Lemma 6.2]. The characteristic

time s(t) is precisely the first positive time such that 〈V (s(t), t), Z〉D = 0. Here,

with a slight abuse of notation, 〈·, ·〉D denotes the scalar product that makes X, Y, Z

orthonormal. The following computation is crucial in what follows. We recall that

we are assuming the φ-curvature to be a constant h 6= 0.

Lemma 7.7. We have the identity

〈V (t, s), Z〉D = 2[h−2ω(Ξ, Ξ) + ω(Ξ− h−1Ξ, h−1µ(τ + hs))

].

Proof. First notice that

〈V, Z〉D = zt + ω(ξt, ξ), (7.19)

where

zt(t, s) = zt(t, 0) +

∫ s

0

ω(ξt(t, σ), ξs(t, σ)) dσ +

∫ s

0

ω(ξ(t, σ), ξst(t, σ)) dσ.

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28 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Using (7.12), (7.13), and the skew-symmetry of ω, the above implies

zt(·, s) = ω(Ξ, Ξ) +

∫ s

0

ω(Ξ− h−1Ξ + h−1τ µ(τ + hσ), µ(τ + hσ)) dσ

+

∫ s

0

ω(Ξ− h−1Ξ + h−1µ(τ + hσ), τ µ(τ + hσ)) dσ

= ω(Ξ, Ξ) + h−1ω(Ξ− h−1Ξ, µ(τ + hs)− µ(τ))

+ h−1ω(Ξ− h−1Ξ, τ µ(τ + hs)− τ µ(τ))

+ h−2ω(µ(τ + hs), τ µ(τ + hs))− h−2ω(µ(τ), τ µ(τ))

= ω(Ξ, Ξ) + h−1ω(Ξ− h−1Ξ, µ(τ + hs))− h−1ω(Ξ− h−1Ξ, Ξ)

+ h−1ω(Ξ− h−1Ξ, τ µ(τ + hs))− h−1ω(Ξ− h−1Ξ, Ξ)

+ h−2ω(µ(τ + hs), τ µ(τ + hs))− h−2ω(Ξ, Ξ)

= ω(Ξ, Ξ)− h−1ω(Ξ, Ξ) + h−2ω(Ξ, Ξ) + ω(Ξ− h−1Ξ, h−1µ(τ + hs))

+ h−1ω(Ξ− h−1Ξ + h−1µ(τ + hs), τ µ(τ + hs)).

Moreover, we have

ω(ξt, ξ) = ω(Ξ− h−1Ξ + h−1τ µ(τ + hs),Ξ− h−1Ξ + h−1µ(τ + hs))

= h−1ω(τ µ(τ + hs),Ξ− h−1Ξ + h−1µ(τ + hs)) + ω(Ξ,Ξ)

− h−1ω(Ξ,Ξ) + ω(Ξ− h−1Ξ, h−1µ(τ + hs)) + h−2ω(Ξ, Ξ).

Summing up, we obtain the claim. �

We show next that for every t ∈ I, the Legendre curve s 7→ γ(t, s) meets a charac-

teristic point before that ξ(t, s) comes back to the point ξ(t, 0) = Ξ(t), i.e., hs(t) < M .

Lemma 7.8. For any t ∈ I, there exists s(t) ∈ (0,M/h) such that 〈V (t, s(t)), Z〉D =

0.

Proof. For fixed t, consider the function θ : [0,M ]→ R, defined by

θ(s) = ω(Ξ− h−1Ξ, h−1µ(τ + hs)).

By Lemma 7.7, we have that 〈V (t, s), Z〉D = 0 if and only if θ(s) = b with b :=

h−2ω(Ξ, Ξ). The equation θ(s) = b is certainly satisfied for hs = nM , n ∈ N. This

follows by the M -periodicity of µ and the fact that V (t, 0) = Γ(t) is horizontal.

It is enough to consider the case b ≥ 0, the case b < 0 being analogous. By (7.13)

we have

θ(0) = ω(Ξ− h−1Ξ, µ(τ)) = ω(µ(τ), µ(τ)).

By the fact that Cφ is a convex curve around 0, it follows that θ(0) 6= 0.

If θ(0) > 0 there exists s∗ ∈ (0,M/(2h)) such that θ(s∗) > θ(0) = b. In this case,

by symmetry of Cφ we have µ(τ + h(s∗ + M/(2h))) = −µ(τ + hs∗), thus implying

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 29

θ(s∗ + M/(2h)) = −θ(s∗) < −b ≤ 0. By continuity of θ, we deduce the existence of

s ∈ (0,M/h) satisfying θ(s) = b. We argue in the same way in the case θ(0) < 0. �

We now determine a quantity that remains constant along the Legendre curves

s 7→ γ(t, s).

Proposition 7.9. For any t ∈ I and for all s ∈ [0, s(t)] we have

〈V (t, s), Z〉D + h〈∇φ†(ξs(t, s)), ξt(t, s)〉 = 0. (7.20)

Proof. By (7.19), (7.14) and (7.17), we have

∂s〈V, Z〉D = zts + ω(ξts, ξ) + ω(ξt, ξs) =

∂tω(ξ, ξs) + ω(ξst, ξ) + ω(ξt, ξs)

= ω(ξt, ξs) + ω(ξ, ξst) + ω(ξst, ξ) + ω(ξt, ξs) = 2ω(ξt, ξs). (7.21)

We claim that

h∂

∂s(〈∇φ†(ξs), ξt〉) = 2ω(ξs, ξt). (7.22)

Indeed, by Theorem 6.2 and Remark 6.3, we have

∂s∇φ†(ξs) = H φ†(ξs)ξss =

1

hξ⊥s , (7.23)

and therefore

∂s〈∇φ†(ξs(t, s)), ξt(t, s)〉 =

1

h〈ξ⊥s , ξt〉+ 〈∇φ†(ξs), ξst〉

On differentiating (7.16) w.r.t. t we see that 〈∇φ†(ξs), ξst〉 = 0. This is (7.22).

Summing up (7.21) and (7.22), we deduce that the function Λt(s) = 〈V (t, s), Z〉D +

h〈∇φ†(ξs(t, s)), ξt(t, s)〉 is constant. To conclude the proof it is enough to check that

Λt(0) = 0. On the one hand, we have 〈V (t, 0), Z〉D = 〈Γ(t), Z〉D = 0, since Γ is

horizontal. On the other hand, since ∇φ†(v) = −∇φ∗(v⊥)⊥ for any v 6= 0, using

(7.15) we finally obtain

〈∇φ†(ξs(t, 0)), ξt(t, 0)〉 = −〈∇φ∗(ξs(t, 0)⊥)⊥, Ξ(t)〉 = 0. �

Since the set Γ1 := {γ(t, s(t)) : t ∈ I} is made of characteristic points, it is

either an isolated point or a nontrivial characteristic curve (Lemma 7.1). We will

see in the proof of Theorem 1.1, contained in Section 8.2, that if Γ1 were an isolated

characteristic point, then the same would be true for Γ. We stress that the argument

leading to such a conclusion does not rely on the characterization of Γ provided in

this section. We then have that Γ1 := {γ(t, s(t)) : t ∈ I} is a nontrivial characteristic

curve.

Proposition 7.10. The function t 7→ s(t) is constant.

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30 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Proof. Let t ∈ I. Since 〈V (t, s(t)), Z〉D = 0, the point γ(t, s(t)) is characteristic for

Σ. Then, by Lemma 7.1 and Remark 7.5, Γ1 is a C2 characteristic curve. By the

implicit function theorem, the function t 7→ s(t) is C1-smooth and for t ∈ I we have

Γ1(t) = V (t, s(t)) + s(t)γs(t, s(t)).

The curve Ξ1 obtained by projecting Γ1 on the xy-plane then satisfies

Ξ1(t) = ξt(t, s(t)) + s(t)ξs(t, s(t)).

Since γ(t, s(t)) ∈ C (Σ), by Proposition 7.4, and using the fact that ∇φ†(v) =

−∇φ∗(v⊥)⊥ for any v 6= 0 we have

〈∇φ†(ξs(t, s(t))), Ξ1(t)〉 = −〈∇φ∗(ξs(t, s(t))⊥)⊥, Ξ1(t)〉 = 0.

Therefore we obtain

0 = 〈∇φ†(ξs(t, s(t))), ξt(t, s(t))〉+ s(t)〈∇φ†(ξs(t, s(t))), ξs(t, s(t))〉, (7.24)

where, by Proposition 7.9,

〈∇φ†(ξs(t, s(t))), ξt(t, s(t))〉 = 0,

and moreover, by (7.16),

〈∇φ†(ξs(t, s(t))), ξs(t, s(t)) = φ†(ξs(t, s(t))) = 1.

Equation (7.24) thus implies s = 0, which concludes the proof. �

We are now ready to prove Theorem 7.2.

Proof of Theorem 7.2. Without loss of generality we assume h > 0. By Remark 7.5,

Γ is of class C2 and we denote by I an interval of parametrization of Γ = (Ξ, ζ)

satisfying (7.10). We consider the parametrization γ given by Lemma 7.6. By Propo-

sition 7.10 the characteristic time s(t) is constant on I and we let s(t) = s ∈ R. Since

〈V (t, s), Z〉D = 0, by Lemma 7.7 we thus have

h−2ω(Ξ(t), Ξ(t)) + ω(Ξ(t)− h−1Ξ(t), h−1µ(τ(t) + hs)) = 0.

Using (7.13), the last equation reads

τω(µ(τ), µ(τ)− µ(τ + hs)) = hω(µ(τ + hs), µ(τ)). (7.25)

If the right-hand side is 0 at some t ∈ I, then µ(τ(t)) and µ(τ(t) +hs) are parallel by

definition of ω (cf. (1.2)). Since hs ∈ (0,M) by Lemma 7.8, the only possible choice

is hs = M/2. Plugging such choice into the left-hand side and using the fact that

µ(τ +M/2) = −µ(τ), we obtain

2τω(µ(τ), µ(τ)) = 0 on I.

This implies that τ = 0 on I and therefore that τ is constant on I. By (7.13) we

deduce that Ξ is constant on I implying that Ξ is a straight line.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 31

We are now left to consider the case hs ∈ (0,M), hs 6= M/2, so that ω(µ(τ(t) +

hs), µ(τ(t))) 6= 0 for every t ∈ I. Equation (7.25) reads

τ = f(τ) with f(τ) :=hω(µ(τ + hs), µ(τ))

ω(µ(τ), µ(τ)− µ(τ + hs)).

For the sake of simplicity, assume 0 ∈ I. Notice that f is M/2-periodic and of class

C1 as a function of τ . Hence, given τ0 ∈ R satisfying µ(τ0) = Ξ(0), there is a unique

maximal solution τ to the differential equation with the initial condition τ(0) = τ0.

Since hs ∈ (0,M), hs 6= M/2, we have f(τ) 6= 0, yielding that τ has constant sign.

To fix the ideas, assume that sign(τ) = 1. Then, there exists T0 > 0 such that

τ(T0) = τ0 +M/2. We claim that

τ(t+ T0) = τ(t) +M

2for all t ∈ R. (7.26)

This follows from the fact that τ1(t) := τ(T0 + t) and τ2(t) := τ(t) + M/2 for t ∈ Rsolve the same Cauchy problem τ(t) = f(τ), τ(0) = τ0 + M/2. Then, by (7.26),

M -periodicity of µ, and (7.13), we have for every t ∈ R

Ξ(t+ 2T0) = µ(τ(t+ 2T0)) = µ(τ(t) +M) = µ(τ(t)) = Ξ(t),

i.e., Ξ is 2T0-periodic. This implies that Ξ is also 2T0-periodic. Indeed, for t ∈ R we

have

Ξ(t+ 2T0)− Ξ(t) =

∫ t+2T0

t

Ξ(σ) dσ =

∫ t+2T0

t

µ(τ(σ)) dσ

=

∫ t+T0

t

µ(τ(σ)) dσ +

∫ t+T0

t

µ(τ(σ + T0)) dσ

=

∫ t+T0

t

µ(τ(σ)) dσ −∫ t+T0

t

µ(τ(σ)) dσ = 0,

where we have used again the symmetry of Cφ and (7.26).

We are left to show that Ξ(σ) 6= Ξ(t) for any 0 ≤ σ < t < 2T0. Assume that

Ξ(σ) = Ξ(t) for some 0 ≤ σ < t ≤ 2T0. Then we have 0 =∫ tσ

Ξ(t) dt =∫ tσµ(τ(t)) dt.

Now, letting v := µ(τ(σ)), by the symmetry of Cφ the function

σ 7→∫ σ

σ

〈µ(τ(t), v)〉 dt

is monotone increasing for σ ∈ [σ, σ + T0] and decreasing for σ ∈ [σ + T0, σ + 2T0].

Hence, the equation∫ tσµ(τ(t)) dt = 0 implies σ = 0 and t = 2T0. �

7.3. Characteristic set of isoperimetric sets. In this section we apply the previ-

ous results to the study of the characteristic set of φ-isoperimetric sets. As a Corollary

of Theorem 7.2 we have the following

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32 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Corollary 7.11. Let φ∗ be of class C2 and let E ⊂ H1 be a φ-isoperimetric set of class

C2. Then C (E) consists of isolated points. Moreover, for every p0 = (ξ0, z0) ∈ C (E)

and every f such that p0 ∈ gr(f) ⊂ ∂E, we have rank(JF (ξ0)) = 2.

Proof. By Remark 3.2, we know that ∂E is bounded. Therefore we exclude the

possibility that C (∂E) contains complete (unbounded) lifts of simple curves. �

Lemma 7.12. Let φ∗ be of class C2 and E ⊂ H1 be a φ-isoperimetric set of class

C2. Let p0 ∈ C (E). There exists r > 0 such that for p ∈ ∂E ∩ B(p0, r), p 6= p0, the

maximal horizontal lift of the φ-circle in ∂E through p meets p0.

Proof. The surface ∂E ∩ B(p0, r) is the z-graph of f ∈ C2(D) and p0 = (ξ0, f(ξ0))

with C (f) ∩ {|ξ − ξ0| < r} = {ξ0}. Let Θξ ⊂ D be the maximal φ-circle (integral

curve of F⊥) passing though ξ ∈ D \ {ξ0}. Notice that the radius of Θξ does not

depend on ξ, as it follows from Corollary 5.4. If ξ0 /∈ Θξ, then the normal vector

Nξ = ∇φ∗(F ) is continuously defined on Θξ.

Assume that there exists a sequence of such ξ with ξ → ξ0. By an elementary

compactness argument it follows that there exists a φ-circle Θ passing through ξ0

and there exists a normal N that is continuously defined along Θ and, in particular,

through ξ0. Outside ξ0 we have N = ∇φ∗(F ).

Let b ∈ R2 the unit vector tangent to Θ at ξ0. Then we have

F (ξ0 + tb) = F (ξ0) + tJF (ξ0)b+ o(t) = tJF (ξ0)b+ o(t),

with JF (ξ0)b 6= 0, because JF (ξ0) has rank 2 by Lemma 7.1. Since ∇φ(−v) =

−∇φ(v), for v ∈ R2 \ {0}, it follows that

limt→0+∇φ∗(F (ξ0 + tb)) = ∇φ∗(JF (ξ0)b),

limt→0−

∇φ∗(F (ξ0 + tb)) = −∇φ∗(JF (ξ0)b).

This contradicts the continuity of N along Θ at ξ0. �

8. Classification of φ-isoperimetric sets of class C2

8.1. Construction of φ-bubbles. Let φ be a norm in R2 that we normalize by

φ(1, 0) = 1. For ξ0 ∈ R2 and r > 0, φ-circles are defined in (1.5) and we let the φ-disk

of radius r and center ξ0 be

Dφ(ξ0, r) = {ξ ∈ R2 : φ(ξ − ξ0) < r}.

We also let Cφ(r) = Cφ(0, r), Cφ = Cφ(1) and Dφ(r) = Dφ(0, r), Dφ = Dφ(1).

The circle Cφ is a Lipschitz curve and we denote by L = Lφ > 0 its Euclidean length.

We parametrize Cφ by arc-length through κ ∈ Lip([0, L];R2

)such that κ([0, L]) = Cφ

with initial and end-point κ(0) = κ(L) = (−1, 0). We choose the anti-clockwise

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 33

orientation and we extend κ to the whole R by L-periodicity. Then we have κ ∈Lip(R;R2).

The map ξ : R2 → R2, ξ(t, τ) = κ(t) + κ(τ), is in Lip(R2;R2). We restrict ξ to the

domain

D ={

(t, τ) ∈ R2 : τ ∈ [0, L], t ∈ [τ + L/2, τ + 3L/2]}.

Notice that ξ(τ + L/2, τ) = ξ(τ + 3L/2, τ) = 0 for any τ ∈ [0, L]. We define the

function z ∈ Lip(D),

z(t, τ) =

∫ t

τ+L/2

ω(ξ(s, τ), ξs(s, τ)

)ds. (8.1)

The map Φ : D → R3 defined by Φ = (ξ, z) is Lipschitz continuous. Moreover, Φ is

Ck if φ is Ck.

We define the Lipschitz surface Σφ = Φ(D) ⊂ R3 and call S = Φ(τ + L/2, τ) =

0 ∈ Σφ the south pole of Σφ and N = Φ(τ + 3L/2, τ) = (0, 0, z(τ + 3L/2, τ)) the

north pole. We call the bounded region Eφ ⊂ R3 enclosed by Σφ the φ-bubble. Eφis a topological ball and it is the candidate solution to the φ-isoperimetric problem.

When φ is the Euclidean norm in the plane, the set Eφ is the well-known Pansu’s

ball.

8.2. Classification of φ-isoperimetric sets of class C2. We are ready to prove

the main theorem of the paper.

Proof of Theorem 1.1. The set E is bounded and connected, by Remark 3.2. We may

also assume that it is open. It follows from Corollary 5.4 (and from the analogous

result for x-graphs and y-graphs based on Remark 5.3) that, out of the characteristic

set C (E), the surface ∂E is foliated by horizontal lifts of φ-circles. Then C (E)

contains at least one point, since otherwise, ∂E would contain an unbounded curve,

contradicting the boundedness of E.

Let f ∈ C2(D), with D ⊂ R2 open, be a maximal function such that gr(f) ⊂ ∂E

and C (f) 6= ∅. We may assume that 0 ∈ C (f), f(0) = 0 and that E lies above

the graph of f near 0. Around the characteristic point 0, the function f must have

the structure described in Lemma 7.12. It follows that, up to a dilation, we have

gr(f) ⊂ ∂Eφ.

The maximal domain for f must be D = Dφ(2). Otherwise, at each point ξ ∈∂D \ ∂Dφ(2) the space T(ξ,f(ξ))∂E = T(ξ,f(ξ))∂Eφ is not vertical, contradicting the

maximality of D. This shows that the graph of f is the ‘lower hemisphere’ of ∂Eφ.

Up to extending f by continuity to ∂D, we have (ξ, f(ξ)) /∈ C (E) for each ξ ∈ ∂D.

Hence there exists a φ-circle passing through 0 whose horizontal lift stays in ∂E

and passes through (ξ, f(ξ)). The collection of all the maximal extensions of such

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34 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

horizontal lifts completes the upper hemisphere of ∂Eφ, thus implying that ∂Eφ ⊂ ∂E.

Moreover, since ∂E is C2, we deduce that ∂Eφ is a connected component of ∂E.

In conclusion we have proved that ∂E is the finite union of boundaries of φ-bubbles

having the same curvature. By connectedness of E this concludes the proof. �

In general, φ-bubbles are not of class C2 and not even of class C1, e.g., in the case

of a crystalline norm. Even when φ is regular, there may be a loss of regularity at

the poles of Eφ.

8.3. Regularity of φ-bubbles. We first show that φ-bubbles have the same regu-

larity as φ outside the poles.

Lemma 8.1. If φ is strictly convex and of class Ck, for some k ≥ 1, then the set

Σφ \ {S,N} is an embedded surface of class Ck.

Proof. If the Jacobian of Φ has rank 2 at the point (t, τ) ∈ D, then Σφ is an em-

bedded surface of class Ck around the point Φ(t, τ). A sufficient condition for this is

det Jξ(t, τ) 6= 0. The Jacobian of ξ : D → R2 satisfies

det Jξ(t, τ) = 0 if and only if κ(t) = ±κ(τ).

The case κ(t) = −κ(τ) is equivalent to κ(t) = −κ(τ), by the strict convexity of the

norm. This is in turn equivalent to t = τ + L/2 or t = τ + 3L/2. In the former case

we have Φ(t, τ) = S, in the latter Φ(t, τ) = N .

We are left to consider the case κ(t) = κ(τ). By strict convexity of φ, this implies

κ(t) = κ(τ), that is equivalent to t = τ + L. In this case, we have ξ(t, τ) = 2κ(τ) ∈Cφ(2) The point Φ(t, τ) is on the ‘equator’ of Σφ.

We study the regularity of Σφ at points Φ(τ + L, τ). The height z(τ + L, τ) does

not depend on τ because it is half the area of the disk Dφ. It follows that 0 =

∂τ(z(τ + L, τ)

)= zt(τ + L, τ) + zτ (τ + L, τ) and this implies that

zt(τ + L, τ) 6= zτ (τ + L, τ), (8.2)

as soon as we prove that the left-hand side does not vanish. Indeed, differentiating

(8.1) we obtain

zt(τ + L, τ) = 2ω(κ(τ), κ(τ)

)6= 0,

because κ(τ) and κ(τ) are not proportional.

From κ(τ + L) = κ(τ) 6= 0 and (8.2), we deduce that the Jacobian matrix JΦ(τ +

L, τ) has rank 2. This shows that Σφ is of class Ck also around the ‘equator’. �

The regularity of Σφ at the poles is much more subtle. We study the problem in

Theorem 1.3, whose proof is presented below.

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 35

Proof of Theorem 1.3. We study the regularity at the south pole. By Lemma 8.1

there exists a function f ∈ C2(Dφ(2) \ {0}) such that the graph of f is the lower

hemisphere of Σφ without the south pole. We shall show that f can be extended to

a function f ∈ C2(Dφ(2)) satisfying ∇f(0) = 0 and Hf(0) = 0. Here and in the

sequel, we denote by Hf the Hessian matrix of f . Differentiating the identity

z(t, τ) = f(ξ(t, τ)), τ ∈ [0, L], t ∈ (τ + L/2, τ + L),

we find the identities

zt(t, τ) = 〈∇f, κ(t)〉, (8.3)

zτ (t, τ) = 〈∇f, κ(τ)〉, (8.4)

ztt(t, τ) = 〈Hfκ(t), κ(t)〉+ 〈∇f, κ(t)〉, (8.5)

zττ (t, τ) = 〈Hfκ(τ), κ(τ)〉+ 〈∇f, κ(τ)〉, (8.6)

ztτ (t, τ) = zτt(t, τ) = 〈Hfκ(t), κ(τ)〉, (8.7)

where, above and in the following, Hf and ∇f are evaluated at ξ(t, τ).

On the other hand, from (8.1) we compute the derivatives

zt(t, τ) = ω(κ(t) + κ(τ), κ(t)), (8.8)

zτ (t, τ) = ω(κ(τ), κ(t) + κ(τ)), (8.9)

ztt(t, τ) = ω(κ(t) + κ(τ), κ(t)), (8.10)

zττ (t, τ) = ω(κ(τ), κ(t) + κ(τ)), (8.11)

ztτ (t, τ) = ω(κ(τ), κ(t)). (8.12)

In formulas (8.3)–(8.12), we will replace κ(τ), κ(τ), and κ(τ) with their Taylor ex-

pansions at the point t− L/2.

By assumption, the arc-length parameterization of the circle Cφ satisfies κ ∈C4(R;R2) and

κ(t) = λ(t)κ(t)⊥, t ∈ [0, L], (8.13)

for a function (the curvature) λ ∈ C2(R) that is L-periodic and strictly positive. So

there exist 0 < λ0 ≤ Λ0 <∞ such that

0 < λ0 ≤ λ ≤ Λ0, |λ| ≤ Λ0, |λ| ≤ Λ0.

The third and fourth derivatives of κ have the representation:

κ(3) = λκ⊥ − λ2κ and κ(4) = (λ− λ3)κ⊥ − 3λλκ. (8.14)

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36 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

In the following, we let δ = t− τ − L/2 > 0. The third order Taylor expansion for

κ(τ) at t− L/2 is

κ(τ) = κ(t− L/2)− δκ(t− L/2) +δ2

2κ(t− L/2)− δ3

6κ(3)(t− L/2) + o(δ3)

= −κ(t) + δκ(t)− δ2

2κ(t) +

δ3

6κ(3)(t) + o(δ3)

= −κ(t) + δκ(t)− δ2

2λ(t)κ(t)⊥ +

δ3

6(λ(t)κ(t)⊥ − λ(t)2κ(t)) + o(δ3). (8.15)

Hereafter, when not explicit, the functions κ and λ and their derivatives are evaluated

at t. The little-o remainders are uniform with respect to the base point t− L/2. By

a similar computation, using (8.14) we also obtain

κ(τ) = −κ+ δλκ⊥ − δ2

2

(λκ⊥ − λ2κ

)+δ3

6(λ− λ3)κ⊥ − δ3

2λλκ+ o(δ3), (8.16)

κ(τ) = −λκ⊥ + δ(λκ⊥ − λ2κ)− δ2

2

((λ− λ3)κ⊥ − 3λλκ

)+ o(δ2). (8.17)

We are ready to start the proof. We will use the identities

ω(κ, κ⊥) = −ω(κ⊥, κ) =1

2. (8.18)

Recall our notation δ = t− τ − L/2.

Step 1. We claim that there exists C > 0 such that

|∇f(ξ(t, τ))| ≤ Cδ2 for all τ ∈ [0, L], t ∈ (τ + L/2, τ + L). (8.19)

This estimate implies that f can be extended to a function f ∈ C1(Dφ(2)) satisfying

∇f(0) = 0.

Inserting (8.15) and (8.16) into (8.8) and (8.9) yields

zt(t, τ) = ω((− δ2

2λ+ δ3

6λ)κ(t)⊥, κ(t)

)=δ2

4λ− δ3

12λ+ o(δ3), (8.20)

zτ (t, τ) = ω((−1 + δ2

2λ2)κ, (− δ2

2λ+ δ3

6λ)κ⊥

)+ ω

((δλ− δ2

2λ)κ⊥, (δ − δ3

6λ2)κ

)+ o(δ3)

= −δ2

4λ+

δ3

6λ+ o(δ3). (8.21)

Now, plugging (8.16) and (8.21) into (8.4) and then using (8.3) and (8.20) we obtain

−δ2

4λ+

δ3

6λ = 〈∇f, κ(t)〉

(−1 + δ2

2λ2− δ3

2λλ)

+ 〈∇f, κ(t)⊥〉(δλ− δ2

2λ+ δ3

6(λ− λ3)

)+ o(δ3)

= −δ2

4λ+

δ3

12λ+ 〈∇f, κ(t)⊥〉

(δλ− δ2

2λ+ δ3

6(λ− λ3)

)+ o(δ3).

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 37

Dividing the last equation by λδ > 0, we get

〈∇f, κ⊥〉 =δ2

12

λ

λ+ o(δ2). (8.22)

Thus, there exists C > 0 such that∣∣〈∇f, κ⊥〉∣∣ ≤ Cδ2.

On the other hand, by (8.3) and (8.20), possibly changing C > 0 we also obtain∣∣〈∇f, κ〉∣∣ = |zt(t, τ)| ≤ Cδ2,

thus yielding (8.19).

Step 2. We claim that the norm of the Hessian matrix Hf satisfies

|Hf(ξ(t, τ))| = o(1) for τ ∈ [0, L], t ∈ (τ + L/2, τ + L), (8.23)

where o(1) → 0 as δ = t − τ − L/2 → 0. This implies that f can be extended to a

function f ∈ C2(Dφ(2)) satisfying Hf(0) = 0.

Plugging (8.10) and (8.15) into (8.5), and then using (8.13), (8.18), and (8.22)

yields

〈Hfκ, κ〉 = ztt(t, τ)− 〈∇f, k〉 = ω(κ(t) + κ(τ), κ

)− 〈∇f, κ〉

= ω((δ − δ3

6λ2)κ+ (− δ2

2λ+ δ3

6λ)κ⊥, λκ⊥

)− 〈∇f, λκ⊥〉+ o(δ3)

2λ− δ2

12λ+ o(δ2). (8.24)

On the other hand, plugging (8.16) into (8.7), and then using (8.24) we get

ztτ (t, τ) = 〈Hfκ(t), κ(t)〉(−1 +

δ2

2λ2

)+ 〈Hfκ, κ⊥〉

(δλ− δ

)+ o(δ2)

= −δ2λ+

δ2

12λ+ 〈Hfκ, κ⊥〉

(δλ− δ

)+ o(δ2),

while from (8.12), (8.16) and (8.18) we get

ztτ (t, τ) = −1

2

(δλ− δ2

)+ o(δ2).

Therefore we obtain the identity(δλ− δ2

)〈Hfκ, κ⊥〉 = ztτ (t, τ) +

δ

2λ− δ2

12λ+ o(δ2)

= −1

2

(δλ− δ2

2λ)

2λ− δ2

12λ+ o(δ2)

=δ2

6λ+ o(δ2),

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38 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

and dividing by λδ > 0 we get

〈Hfκ, κ⊥〉 =δ

6

λ

λ+ o(δ). (8.25)

By symmetry of the Hessian matrix, we also have

〈Hfκ⊥, κ〉 =δ

6

λ

λ+ o(δ). (8.26)

We are left to estimate 〈Hfκ⊥, κ⊥〉. By (8.17), (8.22), (8.3), and (8.20) we obtain

〈∇f, κ(τ)〉 = (−λ+ δλ− δ2

2(λ− λ3)〈∇f, κ⊥〉+ (−δλ2 + 3

2δ2λλ)〈∇f, κ〉+ o(δ2)

= (−λ+ δλ− δ2

2(λ− λ3)) δ

2

12λλ

+ (−δλ2 + 32δ2λλ) δ

2

4λ+ o(δ2)

= − δ2

12λ+ o(δ2). (8.27)

On the other hand, by (8.16), (8.24), (8.25), (8.26) we have

〈Hfκ(τ), κ(τ)〉 = (−1 + δ2

2λ2)2〈Hfκ, κ〉+ (δλ− δ2

2λ)2〈Hfκ⊥, κ⊥)〉

+ 2(−1 + δ2

2λ2)(δλ− δ2

2λ)〈Hfκ, κ⊥〉+ o(δ2)

= (−1 + δ2

2λ2)2( δ

2λ− δ2

12λ+ o(δ2)) + (δλ− δ2

2λ)2〈Hfκ⊥, κ⊥〉

+ 2(−1 + δ2

2λ2)(δλ− δ2

2λ)( δ

6λλ

+ o(δ))

2λ− 5

12δ2λ+ δ2λ2〈Hfκ⊥, κ⊥〉+ o(δ2). (8.28)

Plugging (8.27) and (8.28) into (8.6) we get

zττ (t, τ) =δ

2λ− 5

12δ2λ+ δ2λ2〈Hfκ⊥, κ⊥〉 − δ2

12λ+ o(δ2)

2λ− δ2

2λ+ δ2λ2〈Hfκ⊥, κ⊥〉+ o(δ2).

(8.29)

Moreover, plugging (8.15) and (8.17) into (8.11), and using (8.18), we get

zττ (t, τ) = ω(− δλ2κ+ (−λ+ δλ)κ⊥, δκ− δ2

2λκ⊥

)+ o(δ2) = −δ

2(−λ+ δλ) + o(δ2).

(8.30)

Comparing (8.29) and (8.30) we therefore obtain

δ2λ2〈Hfκ⊥, κ⊥〉 = −δ2λ+

δ2

2λ− δ

2(−λ+ δλ) + o(δ2) = o(δ2).

This yields 〈Hfκ⊥, κ⊥〉 = o(1) as δ → 0. Together with (8.24), (8.25), and (8.26)

this implies (8.23) and concludes the proof of the theorem. �

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THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 39

9. The isoperimetric problem for general norms

In the case of crystalline norms, the first order necessary conditions satisfied by

an isoperimetric set are not sufficient to reconstruct its structure, even assuming

sufficient regularity. In this section we show that the φ-isoperimetric problem for a

general norm – in particular for a crystalline norm – can be approximated by the

isoperimetric problem for smooth norms.

By Theorem 1.3, we know that if φ is of class C∞+ then the φ-bubble Eφ is of

class C2. In this section, we show that the validity of Conjecture 1.4 implies the

φ-isoperimetric property for the φ-bubble of any (crystalline) norm.

9.1. Smooth approximation of norms in the plane. We start with the mollifi-

cation of a norm.

Proposition 9.1. Let φ be a norm in R2. Then, for any ε > 0 there exists a norm

φε of class C∞+ with dual norm of class C∞, such that for all ξ ∈ R2 we have

(1− η(ε))φε(ξ) ≤ φ(ξ) ≤ (1 + η(ε))φε(ξ), (9.1)

and η(ε)→ 0 as ε→ 0+.

Proof. For ε > 0, we introduce the smooth mollifiers %ε : R → R, supported in

[−επ, επ] defined by

%ε(t) =

cε exp(

π2ε2

t2−π2ε2

)if |t| < π,

0 if |t| ≥ π,

where cε is chosen in such a way that∫R %ε(t) dt = 1. Following [9, 21], we define the

function ψε : R2 → [0,∞) letting

ψε(ξ) :=

∫R%ε(t)φ(Rtξ) dt,

where Rt denotes the anti-clockwise rotation matrix of angle t. The function ψε is a

C∞ norm. On the circle S1 = {ξ ∈ R2 : |ξ| = 1}, the norms ψε converge uniformly

to φ as ε→ 0+. So our claim (9.1) with η(ε)→ 0 holds with ψε replacing φε, by the

positive 1-homogeneity of norms.

We let φε : R2 → [0,∞) be defined by

φε(ξ) :=√ψε(ξ)2 + ε|ξ|2, ξ ∈ R2.

This is a C∞ norm in R2 and (9.1) is satisfied with η(ε) → 0. The unit φε-circle

centered at the origin is the 0-level set of the function

Fε(ξ) = ψ2ε(ξ) + ε|ξ|2 − 1, ξ ∈ R2.

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40 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

Since the Hessian matrix of the squared Euclidean norm is proportional to the identity

matrix I2 and ψ2ε is convex, we have that HFε ≥ 2εI2 in the sense of matrices. Then

the curvature λε of a unit φε-circle satisfies

λε =〈H Fε∇F⊥ε ,∇F⊥ε 〉

|∇Fε|3≥ 2ε

|∇Fε|> 0.

The proof that the dual norm of a norm of class C∞+ is itself of class C∞ is standard

and we omit it. �

9.2. Crystalline φ-bubbles as limits of smooth isoperimetric sets. Let φ be

any norm in R2 and let {φε}ε>0 be the smooth approximating norms found in Propo-

sition 9.1.

Given a Lebesgue measurable set F ⊂ R2, from (9.1) and from the definition of

perimeter (Definition 2.1), we have

(1− η(ε))Pφ(F ) ≤Pφε(F ) ≤ (1 + η(ε))Pφ(F ). (9.2)

The φε-circles Cφε converge in Hausdorff distance to the circle Cφ. This implies

that the φε-bubbles Eφε converge in the Hausdorff distance to the limit bubble Eφ.

This in turn implies the convergence in L1(H1), namely,

limε→0+

L 3(Eφε∆Eφ) = 0, (9.3)

where ∆ denotes the symmetric difference of sets.

Proof of Theorem 1.5. Let F ⊂ H1 be any Lebesgue measurable set with 0 < L 3(F ) <

∞. Assuming the validity of Conjecture 6.4, Eφε is isoperimetric for any ε > 0. So

using twice (9.2) we find

Isopφ(F ) ≥ Isopφε(F )

1 + η(ε)≥ Isopφε(Eφε)

1 + η(ε)≥ 1− η(ε)

1 + η(ε)Isopφ(Eφε).

By the lower semicontinuity of the perimeter with respect to the L1 convergence and

from (9.3), we deduce that

lim infε→0+

Isopφ(Eφε) ≥ Isopφ(Eφ),

and using the fact that η(ε)→ 0 we conclude that Isopφ(F ) ≥ Isopφ(Eφ). �

References

[1] L. Ambrosio. Some fine properties of sets of finite perimeter in Ahlfors regular metric measure

spaces. Adv. Math., 159(1):51–67, 2001.

[2] A. A. Ardentov, E. Le Donne, and Y. L. Sachkov. Sub-Finsler geodesics on the Cartan group.

Regul. Chaotic Dyn., 24(1):36–60, 2019.

[3] J.-P. Aubin and H. Frankowska. Set-valued analysis. Modern Birkhauser Classics. Birkhauser

Boston, Inc., Boston, MA, 2009. Reprint of the 1990 edition [MR1048347].

Page 41: arXiv:2007.11384v1 [math.DG] 22 Jul 2020 · THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 VALENTINA FRANCESCHI1, ROBERTO MONTI2, ALBERTO RIGHINI2, AND MARIO SIGALOTTI1

THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 41

[4] Z. M. Balogh. Size of characteristic sets and functions with prescribed gradient. J. Reine Angew.

Math., 564:63–83, 2003.

[5] D. Barilari, U. Boscain, E. Le Donne, and M. Sigalotti. Sub-Finsler structures from the time-

optimal control viewpoint for some nilpotent distributions. J. Dyn. Control Syst., 23(3):547–575,

2017.

[6] V. N. Berestovskiı. Geodesics of nonholonomic left-invariant inner metrics on the Heisenberg

group and isoperimetrics of the Minkowski plane. Sibirsk. Mat. Zh., 35(1):3–11, i, 1994.

[7] L. Capogna, D. Danielli, and N. Garofalo. The geometric Sobolev embedding for vector fields

and the isoperimetric inequality. Comm. Anal. Geom., 2(2):203–215, 1994.

[8] J.-H. Cheng, J.-F. Hwang, A. Malchiodi, and P. Yang. Minimal surfaces in pseudohermitian

geometry. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4(1):129–177, 2005.

[9] W. P. Dayawansa and C. F. Martin. A converse Lyapunov theorem for a class of dynamical

systems which undergo switching. IEEE Trans. Automat. Control, 44(4):751–760, 1999.

[10] A. Figalli, F. Maggi, and A. Pratelli. A mass transportation approach to quantitative isoperi-

metric inequalities. Invent. Math., 182(1):167–211, 2010.

[11] I. Fonseca. The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A, 432(1884):125–145,

1991.

[12] I. Fonseca and S. Muller. A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh

Sect. A, 119(1-2):125–136, 1991.

[13] V. Franceschi, G. P. Leonardi, and R. Monti. Quantitative isoperimetric inequalities in Hn.

Calc. Var. Partial Differential Equations, 54(3):3229–3239, 2015.

[14] V. Franceschi, F. Montefalcone, and R. Monti. CMC spheres in the Heisenberg group. Anal.

Geom. Metr. Spaces, 7(1):109–129, 2019.

[15] V. Franceschi and R. Monti. Isoperimetric problem in H-type groups and Grushin spaces. Rev.

Mat. Iberoam., 32(4):1227–1258, 2016.

[16] B. Franchi, R. Serapioni, and F. Serra Cassano. Meyers-Serrin type theorems and relaxation of

variational integrals depending on vector fields. Houston J. Math., 22(4):859–890, 1996.

[17] N. Garofalo and D.-M. Nhieu. Isoperimetric and Sobolev inequalities for Carnot-Caratheodory

spaces and the existence of minimal surfaces. Comm. Pure Appl. Math., 49(10):1081–1144, 1996.

[18] K. Lange. MM optimization algorithms. Society for Industrial and Applied Mathematics,

Philadelphia, PA, 2016.

[19] G. P. Leonardi and S. Rigot. Isoperimetric sets on Carnot groups. Houston J. Math., 29(3):609–

637, 2003.

[20] L. V. Lokutsievskiı. Convex trigonometry with applications to sub-Finsler geometry. Mat. Sb.,

210(8):120–148, 2019.

[21] P. Mason, U. Boscain, and Y. Chitour. Common polynomial Lyapunov functions for linear

switched systems. SIAM J. Control Optim., 45(1):226–245, 2006.

[22] R. Monti. Heisenberg isoperimetric problem. The axial case. Adv. Calc. Var., 1(1):93–121, 2008.

[23] R. Monti and M. Rickly. Convex isoperimetric sets in the Heisenberg group. Ann. Sc. Norm.

Super. Pisa Cl. Sci. (5), 8(2):391–415, 2009.

[24] P. Pansu. Une inegalite isoperimetrique sur le groupe de Heisenberg. C. R. Acad. Sci. Paris

Ser. I Math., 295(2):127–130, 1982.

[25] J. Pozuelo and M. Ritore. Pansu-Wulff shapes in H1. arXiv e-prints, page arXiv:2007.04683,

July 2020.

Page 42: arXiv:2007.11384v1 [math.DG] 22 Jul 2020 · THE ISOPERIMETRIC PROBLEM FOR REGULAR AND CRYSTALLINE NORMS IN H1 VALENTINA FRANCESCHI1, ROBERTO MONTI2, ALBERTO RIGHINI2, AND MARIO SIGALOTTI1

42 V. FRANCESCHI, R. MONTI, A. RIGHINI, AND M. SIGALOTTI

[26] M. Ritore. A proof by calibration of an isoperimetric inequality in the Heisenberg group Hn.

Calc. Var. Partial Differential Equations, 44(1-2):47–60, 2012.

[27] M. Ritore and C. Rosales. Area-stationary surfaces in the Heisenberg group H1. Adv. Math.,

219(2):633–671, 2008.

[28] R. T. Rockafellar. Convex analysis. Princeton Landmarks in Mathematics. Princeton University

Press, Princeton, NJ, 1997. Reprint of the 1970 original, Princeton Paperbacks.

[29] Y. L. Sachkov. Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems. Regul. Chaotic

Dyn., 25(1):33–39, 2020.

[30] A. P. Sanchez. A Theory of Sub-Finsler Surface Area in the Heisenberg Group. PhD thesis,

Tufts University, 2017.

[31] G. Wulff. Zur Frage der Geschwindigkeit des Wachsthums und der Auflosung der Krys-

tallflachen. Z. Kristallogr., 34:449–530, 1901.

1Laboratoire Jacques-Louis Lions, Sorbonne Universite, Universite de Paris, Inria,

CNRS, Paris, France

E-mail address: [email protected]

E-mail address: [email protected]

2Dipartimento di Matematica Tullio Levi Civita, Universita di Padova, Italy

E-mail address: [email protected]

E-mail address: [email protected]